Rida Ahmed Ammar
email: rammar31@yahoo.com

Mostefa Lemerini

Surface plasmon polariton in metal-insulator-metal configuration

Keywords: Drude-Lorentz, surface Plasmon polariton, wave propagation, optical waveguides

The optics of the surface plasmon resonance has been known for a long time. In the configuration multilayer, the optical coupling of a wave incident to collective oscillations of electrons along an interface between a metal and a dielectric is governed by the thickness of metal and gap layers. The surface plasmon excitations excited by an electromagnetic wave in the visible band (λ= 633 nm). For the metal, in particular a frequency on their dielectric permittivity dependence and described by the Drude-Lorentz model and using the effectiveindex approach and an explicit expression for the propagation constant of gap surface plasmon polaritons (G-SPPs) obtained for moderate gap widths.

Introduction

Surface plasmon polaritons (SPP) are electromagnetic excitations propagating at the interface between a dielectric and a conductor (usually metal) material possessing opposite signs of the real part of their dielectric permittivities, evanescently confined in the perpendicular direction. These electromagnetic surface waves arise via the coupling of the electromagnetic fields to oscillations of the conductor's electron plasma. Taking the wave equation as a starting point, this section describes the fundamentals of surface plasmon polaritons both in metal /dielectric multilayer structures (IMI and semi-infinite MIM waveguides) [START_REF] Han | Radiation guiding with surface plasmon polaritons[END_REF][START_REF] Homola | Electromagnetic Theory of Surface Plasmons[END_REF].

The TM polarized SP mode is uniquely characterized by its magnetic field lying in the plane of the metal-insulator surface and perpendicular to the wave propagation direction. The metal commonly used to excite surface plasmon polaritons (SPPs) is silver (Ag) due to their remarkable optical properties described by the frequency dependent complex permittivity ε ω ε ω i ε ω in the Drude-Lorentz model ε 0, |ε | ≫ ε . Since the SPR is the resonance phenomenon corresponding to an energetic transfer from incident light to SPPs.

The E7 [START_REF] Li | Refractive Indices of Liquid Crystals for Display Applications[END_REF][START_REF] Li | High temperature-gradient refractive index liquid crystals[END_REF], nematic liquid crystals mixture contains cyanobiphenyl and cyanoterphenol components, at a specific composition, which possess relatively high birefringence and positive dielectric anisotropy. Due to these properties, it is widely used in polymer dispersed liquid crystals [START_REF] Bedjaoui | Preferential salvation of the eutectic mixture of liquid crystal E7 in a polysiloxane[END_REF]. The specific composition is critical to ensure physical properties and characteristic of the liquid crystal. The specific composition is critical to ensure physical properties and characteristic of the liquid crystal. Even small changes can have pronounced effects on factors such as the nematic to isotropic transition, and glass transition temperatures. E7 is used as the dielectric ε in the temperature 25 c on MIM configuration. We wish to clarify that we are working under conditions of plasmon wave excite the surface polariton by an electromagnetic wave in the act of taking the negative real part of the dielectric permittivity of the metal. The characteristic parameters of the guide are the depth of propagation of electromagnetic waves in the metal medium, the permittivity of the metal and the middle of the opening, the dimensions (length and width) of the guide, the relationship of the dispersion depends mainly the effective index, and the cutoff frequency. In this work, we developed a geometry of the waveguides for confining propagating resonant modes. The main feature of the dimensions is the wavelength sub-length scale with respect to the wavelength λ of the excitation wave. In their simplest form, the light guides consist of layered material media. For that light energy can be propagated and confined through the guide, the indices of these media are recorded by optical usual conditions [START_REF] Sonia | Theoretical study of loss compensation in long-range dielectric loaded surface plasmon polariton waveguides[END_REF][START_REF] Kou | Multimode interference demultiplexers and splitters in metal-insulator-metal waveguides[END_REF].

The Drude-Lorentz model:

Drude-Lorentz model is often used for parameterization of the optical constants of metals. In addition to the conduction electrons, the Drude-Lorentz model takes into account the bound electrons. The interband transition of electrons from filled bands to the conduction band can significantly influence the optical response. In alkali metals, these transitions occur at high frequencies and provide only small corrections to the dielectric function in the optical domain. These metals are well described by the Drude model. On the other side, in noble metals a correction must be made to the dielectric function. It is due to transitions between the bands d and the conduction band s-p. The contribution of bound electrons to the dielectric function can be described by the Lorentz model. To the above Drude dielectric function, a Lorentzian term is added [START_REF] Baida | Finite Difference Time Domain Method for Grating Structures, Gratings: Theory and Numeric Applications[END_REF]:

ε ω ε ω ε ω (1)
Vial et al. [START_REF] Vial | Improved analytical fit of gold dispersion: application to the modeling of extinction spectra with a finite-difference time-domain method[END_REF] suggested a single oscillator leading to a single Lorentzian additional term to well describe the permittivity of gold in the optical range compared with the classical Drude model. In this case, the relative dielectric function is [START_REF] Rakic´ | Optical properties of metallic films for vertical-cavity optoelectronic devices[END_REF]:

ε ω 1 ω ω ω Γ ∑ ω Ω ω ωΓ ( 2
)
where w is the plasma frequency, k is the number of oscillators with frequency Ω , strength f , and life time 1 Γ .

Table .1: Optimized parameters of the Drude-Lorentz model for Silver metal. [START_REF] Li | High temperature-gradient refractive index liquid crystals[END_REF], ω p , Ω j and Γ j are in electron volts, f j has no units.

In Fig. 1, we have plotted the real and imaginary parts of the dielectric function of silver as tabulated in [START_REF] Rakic´ | Optical properties of metallic films for vertical-cavity optoelectronic devices[END_REF], as well as the description achieved using the DL model. The metal has a negative real component of the permittivity in the visible and infrared wavelengths but at shorter wavelengths ε r becomes positive.

Surface plasmons in MIM-Structure (Metal Gap):

An important extension of the simple metal surface is a three layer system, sometimes also called heterostructure [START_REF] Prade | Guided waves in planar heterostructures with negative dielectric constant[END_REF], where each of the layers has an infinite extension in two dimensions. Two basic heterostructures can be distinguished, a dielectric gap in a metal, or MIM (metal-insulator-metal) system and a metal film surrounded by two dielectrics, or IMI (insulator-metal-insulator) system [START_REF] Economou | Surface plasmons in thin films[END_REF]9]. Dionne and colleagues [START_REF] Dionne | Plasmon slot waveguides: Towards chip-scale propagation with sub wavelength-scale localization[END_REF] extended the investigation to SP's propagation in a Ag-SiO 2 -Ag plasmonic (slot) waveguide and determined dispersion relation, energy density and localization for the symmetric and anti-symmetric gap plasmons with respect to the middle plane. One advantage of a MIMstructure is the strong confinement of the field in the dielectric, which is due to a small penetration depth (~ 20 nm) into the metal on each side of the gap. For the same structure, the propagation lengths of 10 μm are supported with localization of the field in the gap of ~20 nm. Dionne et al. [START_REF] Dionne | Plasmon slot waveguides: Towards chip-scale propagation with sub wavelength-scale localization[END_REF][START_REF] Isaac | Surface plasmon mediated transmission of subwavelength slits at THz frequencies[END_REF] also pointed out the additional benefit of those structures for simultaneous use with conventional electronic devices. We consider the gap is the E7 nematic liquid crystal. Fig. 2 shows the refractive index of E7 as a function of wavelength between 400 and 650 nm, which refractive index decrease with the increasing of wavelength. We consider the dielectric is the E7 nematic liquid crystal at T=25 c °. The complex dielectric function ε and the complex index of refraction n are defined as [START_REF] Ordal | Pt, Ag, Ti, and W in the infrared and far infrared[END_REF]:

_____ ε r --------ε i Wavelength, λ (nm)
ε ε i ε n n i k (3)
At T=25 c ° and λ=633 nm, we have ne=1.7305, n0=1.5189, [START_REF] Li | Refractive Indices of Liquid Crystals for Display Applications[END_REF] where : n n i 0 n 1.58943 and ε ε n 2.5263.

In this section we consider two infinite metal planes separated by a gap filled with a E7 nematic liquid crystal in which a film plasmon propagates constitute a MIM [START_REF] Chen | Coupled-Resonator-Induced Fano Resonances for Plasmonic Sensing with Ultra-High Figure of Merits[END_REF][START_REF] Zhang | Fano Resonance in a Gear-Shaped Nanocavity of the Metal-Insulator-Metal Waveguide[END_REF][START_REF] Liu | High-extinctionratioandlow-insertion-loss Plasmonic Filter with Coherent Coupled Nano-cavity Array in a MIM Waveguide[END_REF][START_REF] Sonia | Theoretical study of loss compensation in long-range dielectric loaded surface plasmon polariton waveguides[END_REF][START_REF] Kou | Multimode interference demultiplexers and splitters in metal-insulator-metal waveguides[END_REF][START_REF] Yu | Integrated Tunable Optofluidics Optical Filter Based on MIM Side-Coupled-Cavity Waveguide[END_REF][START_REF] Wen | Design of Plasmonic Comb-Like Filters Using Loop-Based Resonators[END_REF] (metal-insulator-metal) or simply a slot waveguide. We consider the SPP modes in the symmetric MIM configuration of a thin dielectric layer with the thickness w being sandwiched between two metal surfaces. When two identical SPP modes start overlapping with each other for small layer thicknesses, the dispersion relation for the metal gap of the symmetric field distribution can be written as [START_REF] Han | Radiation guiding with surface plasmon polaritons[END_REF][START_REF] Bozhevolnyi | General properties of slow-plasmon resonant nanostructures: nano-antennas and resonators[END_REF].

tanh k ε ε and k , β ε , k (4)
where β denotes the propagation constant of the fundamental gap surface plasmons polariton (GSPP) mode with the transverse field component Ez, having the same sign across the gap, and ε , are the permttivities of metal and insulator (E7). For sufficiently small gap widths (w→ 0), one can use the approximation tanh(x) ≈ x resulting in the following expression [START_REF] Han | Radiation guiding with surface plasmon polaritons[END_REF][START_REF] Bozhevolnyi | General properties of slow-plasmon resonant nanostructures: nano-antennas and resonators[END_REF]: where Re{} and Im{} denote the real and imaginary parts of a complex number, respectively; the attenuation b is in dBcm -1 if β is given in m -1 .

Fig. 3: Schematic of MIM plasmonic waveguide.

The schematic structure of the Ag-E7-Ag plasmonic waveguide (PW) is shown in Fig. 3. This waveguide consists of a low-index E7 stripe (n E7 = 1.58943) sandwiched between a rectangular a silver film (λ = 633 nm, ε Ag =-14.4688-i 1.09378 [START_REF] Rakic´ | Optical properties of metallic films for vertical-cavity optoelectronic devices[END_REF].) The E7 stripe has dimensions of W nm width. The thickness of the silver film is taken as 150 nm largely to ensure that there are no effects of the silver film thickness on the plasmonic mode inside the E7 stripe. Coupling between the cavity resonator and input waveguide depends on the thickness of the metallic gap between the sketched pieces of Ag metal. Attenuation, b (dB cm -1 ) ×10 6 Width, w (nm)

Results and discussion

Width, w (nm)

Propagation length, Lp (µm) b) c) Fig. 4: The effective index n eff , propagation length Lp, and the attenuation b, as functions of width of the gap of the E7 w for λ=633 nm (MIM plasmonic waveguides Ag/E7/Ag, ε 2.5363 and silver metal ε 14.4688 i 1.09378.)

Fig. 4 shows the mode effective index n eff , the propagation length Lp and the attenuation b of SPP mode as functions of the thickness w of the E7 stripe, where the width w of the E7 was chosen as 633 nm. The effective index, propagation length and attenuation could be acquired by numerically solving Eq. (4). We observed that increasing the width of the E7 stripe reduces the mode effective index and the attenuation, and increases the propagation length. However, for E7 thinner than 400 nm, the fundamental mode turns progressively into a plasmonic wave that propagates along the interface (n eff =1.61), leading to poor vertical confinement.

The SPP dispersion curves for Ag-E7-Ag MIM structures with various E7 layer thicknesses are illustrated in Fig. 5, calculated with the OptiFDTD [24]. The result shows decreasing in film thickness and the MIM symmetric mode exhibits a cut-off for core films. In Fig. 6 (a), the magnetic field profile is presented in the plane (x, z) for λ=633 nm (in vacuum) and for a symmetric MIM configuration with permittivities ε 2.5363 and ε 14.4688 i 1.09378 . Thus, the transverse magnetic field Hy(x), which is continuous at the interface, the gap plasmons can be excited in phase, hence the magnetic field component Hy is symmetric with respect to the centre of the gap. As the operation wavelength increases, the delocalization of the two peaks of the SPP mode in the two metaldielectric interfaces causes more power be concentrated in the E7-core of the gap waveguide. Fig. 6(b) shows the transmission as a function of the wavelength, calculated with the OptiFDTD [24] method for the optimized gap distance of 50 nm and length l = 100. It is observed that the maximum of the transmission is 0.7 µm.

Conclusion

The optics of the surface plasmon resonance has been known for so long. In this paper, we discuss the use of a metal-insulator-metal (MIM) structure to generate plasmon surface polaritons. In the first step, we study the influence of the gap thickness on the resonance SPP. In the second step, we present the analytical results of the effective index, attenuation and propagation length as a function of wavelength for disposal, which are excited by an electromagnetic wave in the visible band (λ=633 nm). We take the gap for E7, which is the nematic liquid crystals mixture. For metal, we took a particular frequency dependence on their dielectric permittivity  Ag (). We finally find the basic characteristics for SPP. Based on the results, we have shown that the guide modes depend on its shape (dimensions). This dependence provides a means to control the propagation length, depending on the thickness.

Fig. 1 :

 1 Fig.1: The real and imaginary components of the permittivity of Ag

Fig. 2 :

 2 Fig. 2: The refractive index of the liquid crystal (E7) as a function of wavelength between 450-656 nm [3].

.

  Here, k 0 is the wave vector of incident light, λ is the wavelength, α represents the GSPP propagation constant in the limit of very narrow gaps (w→ 0). The imaginary part of the propagation constant is associated with the attenuation and propagation length of the surface plasmon in the direction of propagation. The propagation constant is related to the effective index n , propagation length L and attenuation b[START_REF] Homola | Electromagnetic Theory of Surface Plasmons[END_REF][START_REF] Homola | Surface Plasmon Resonance Sensors for Detection of Chemical and Biological Species[END_REF] as:

Fig. 5 :Fig. 6 :

 56 Fig.5: Geometry and characteristic tangential magnetic field profile Hy for the semi-infinite MIM wave guide core insulator thickness w, propagate along the positive Z direction.


P Baur1

A Roy1

P Casari2

D Choqueuse3

P Davies3

Structural mechanical testing of a full-size adhesively bonded motorboat

Keywords: adhesive, structural testing, marine composites, impact, sea trials

This paper describes the tests performed on a full-size motorboat to demonstrate the potential for adhesive bonding to replace overlaminated connections. Adhesive bonding resulted in a significant reduction in assembly time for bulkhead connections compared with overlamination. Drop tests and sea trials were performed using specially adapted strain gauge instrumentation. These indicated low adhesive joint strain levels, even for severe test conditions. No damage was observed. The data obtained will be used to improve laboratory test procedures to simulate service loading of boat structures.

INTRODUCTION

(a) to improve the manufacturing process, (b) to evaluate the loads applied to the connections at sea, Composite materials have been extensively used in the pleasure boat industry for many years [1-3]. The manu-(c) to verify by a drop test the capability of the structure to support severe loading. facture of small pleasure boats (motorboats and sailing boats from about 5 to 8 m long) involves the assembly of different components, the main ones being the deckhull and bulkhead-hull connections. These operations 2 MANUFACTURING OF THE PROTOTYPE are time consuming and can account for up to 30 per cent of the total manufacturing time. In order to reduce

The structure retained for this study is a 5.75 m long motorboat designed for sea fishing and excursions this time and the associated labour costs, the use of bonding is now being considered by some boatyards to (Fig. 1a). The characteristics of the boat are shown in Table 1. replace the overlaminating method (which is currently widely used). This boat is an improved standard boat, and the manufacturing process has been modified in order to To validate this manufacturing option, a research programme including research centres, adhesive manu-replace the overlaminating of the bulkheads onto the hull structure by a bonding operation. Figure 1b shows facturers and boatyards started several years ago. Many of the results from the initial studies, which focused on the interior of the boat before bonding the deck. The adhesive chosen is a filled vinyl ester based the mechanical behaviour and durability of adhesively bonded specimens, have been reported [4][5][6][7]. In parallel adhesive provided by the Reichhold company. Other with tests carried out in the laboratory on subcomponents, an adhesively bonded prototype has been manufactured, The drop in measured failure load is related to an increase in peel loading as the bondline gets thicker. phthalic polyesters were also evaluated in a preliminary test programme [7], but this vinyl ester was retained on There may also be more defects in the thicker joints. Thus, apparent strengths of these adhesives do show a account of its superior long-term behaviour. It should be noted that epoxy-based adhesives were considered to dependence on thickness, and when possible it is preferable to reduce bondline thickness, particularly in highly be too expensive for this application. The adhesive is applied in a very simple way, and no surface preparation loaded regions.

For the bulkhead-hull assembly, the bulkheads were of either the GRP or the plywood was performed prior to bonding. The gap between the bulkhead and the hull placed in position in the boat without fixing, and then a length of adhesive was applied on each side of the varied from less than 1 mm up to over 10 mm. Such variations are not unusual, but adhesive bondline thick-bulkhead and manually smoothed. The time needed for this operation is 45 min, whereas the overlaminating ness can have a strong influence on joint strength, and this was one of the parameters examined in the initial operation requires 4 h. For the bulkhead-deck assembly the adhesive was first deposited on the edge of the ply-test programme. Figure 2 shows an example of results.

wood, and then the deck was put in place without pressure.

In this first phase of the project the bonding process has only been applied to the assembly of the bulkhead with the hull and deck structure. The full bonding including the hull-deck connection will be considered at a later date.

INSTRUMENTATION

Few experiments have been performed to measure the strains during navigation of composite pleasure boats, though there is considerable experience of slamming Three locations were retained for study, based on the strains on a large composite motor cruiser [11]. The boatbuilder's experience and discussions with a naval anisotropy of the materials, the low strain values on architect. These are regions in which joint damage had running sections of the structure, the dynamics of the been observed after severe loading in service, and are signal (acquisition rates of at least 500 Hz are needed), shown in Fig. 4. the limited knowledge of the real geometry (thickness,

The measurements are made using sensors composed resin content, ply orientation, etc.) at a given point and of strain gauge combinations, which are directly bonded the poor stability of strain gauge measurements on low to the adhesive joint of a bulkhead-hull connection. A heat dissipation materials (GRP) combine to make this full Wheatstone bridge is built (four 120 ohm gauges) type of measurement very difficult. Taking into account in order to increase the sensitivity, and the wiring is these parameters, and the aim of the study being to adapted in order to concentrate on the loading of interest. determine the loading of the bonded joint, a special Thus, each instrumented connection allows the measureinstrumentation system was defined.

ment of a particular response by eliminating the other The load applied to motor boats differs from that responses. For example, for the tension-compression applied to sailing vessels. For the latter the transfer of (T-C ) sensor, rosettes of 0-90°gauges are placed on the wind loading by the mast and the rigging has to be each side of the joint and the wiring is chosen in order considered, whereas for motor boats the transfer of the to eliminate the bending and temperature effects (Fig. 5). power generated by the engine (inboard or outboard)

This sensor is located on the main central perpendicular is of prime importance. For all vessels the load induced bulkhead (Fig. 4). For the bending (B) sensor the same by slamming (repeated wave loading) has to be conpositioning of the gauges is performed, but the wiring sidered. However, for the motor boat considered in differs in order to eliminate the tension-compression this study, in order to simplify the mechanical loading applied to the connections, they can be split into (Fig. 3):

effect. For the shear (S) sensor the gauges are placed at the impact response of boat sections [START_REF] Hayman | Slamming drop tests on a GRP sandwich model[END_REF]. Here, the boat was lifted by a crane and dropped onto calm water in the 20 m deep section of the IFREMER test basin in Brest. Series of 2 s data recordings were made during the first phase of contact with the water. Figure 7 shows illustrations of the behaviour of the boat during the drop. Different configurations have been retained for this test. They are reported in Table 2. For the fifth test the sandbags were distributed as follows: 120 kg at the engine position, 400 kg in the cockpit, 80 kg on the aft side of the boat. This distribution simulates the loads in navigation. Some examples of results are presented on the plots in Fig. 8.

The maximum value is obtained during the first 200 ms after the impact with the water, and then the signal is quickly damped to reach a value less than 10 per cent of the maximum value after less than 2 s. A general impacts are reported in Table 3. These values are discussed J4=aDt; R=J1+J3-J2-J4>R=21k1e t further below.

45°to the connection axis (Fig. 6). It should also be noted that the wiring of the shear sensors results in a 5 SEA TRIALS sensitivity twice that of the other sensors.

The bending and shear sensors are placed sym-The sea trials were performed in the Atlantic Ocean off metrically on the back of the two longitudinal bulkheads.

the West coast of France. During navigation the boat An HBM Spider@ acquisition data system coupled to a was loaded with three passengers. The sea conditions portable computer is used to collect the data. Depending were quite good (sea condition level 3-4) with a 30 cm on the type of test, a 2 or 10 s data sequence is recorded.

high wave. The measurements were made at a maximum Data recording is triggered by a signal increase. speed of 30 knots and for different vessel orientations with respect to the wave direction (meeting the wave direction head on, at 45°, and with the wave aft) in both 4 DROP TEST straight-on navigation and during turns. Two examples of results are given in Fig. 9. The first type of test that was performed was a drop test. Similar tests have been used in the past at DNV to study The first point to highlight is that no damage has been test and can be attributed to a general mode shape of observed on the boat after these two series of tests. This the structure.

confirms the potential for adhesive bonding to manu-The maximum values are observed for peaks which facture the joints in motorboats of this type, though have a value 5 times higher than the general value of course a final judgement will only be possible when observed during navigation. This peak level must occur information on long-term behaviour of the structure is at the moment when the boat motion is out of phase available after many years of in-service navigation of with the sea surface.

the boat. During tacking, a significant increase in signal is

In order to establish how close the adhesive is to observed in the bending sensor. However, the level damage and failure, the values recorded by the sensors obtained appears to be much lower than the peak during the trials (Tables 3 and4) should be related to observed during the full-speed navigation. The maxithe corresponding material properties. Unfortunately, mum values observed on each sensor are reported in this is not simple. The values of strain, Dl/l, in percent-Table 4.

ages, have been indicated in these tables. These values It is apparent that these values are considerably lower take into account the gauge factor and the type of bridge than the maximum values recorded in the drop tests.

wiring. However, as the loading conditions cannot be Their significance is discussed further below.

clearly defined, and the joint geometry is not perfect (variable shape and symmetry), it is very hard to make this correlation, so the values noted must be considered sensor indicate that peak values are compressive rather than tensile, and the maximum tensile strain was even concentrations near the joint ends, requires very large models, and the strains in these regions are impossible lower.

Loading rate effects may affect material properties. to measure. Nevertheless, some comments may be made.

The data usually available for an adhesive are quasi-The highest strain rates during the prototype tests are around 0.05 s-1. At hot spots (the ends of the joints) static tensile stress-strain plots. An example is shown in Fig. 10 for the filled vinyl ester adhesive used here.

these could be even higher. Strain rates in tensile tests are typically 50 times slower than this value, so higher The strain at failure is considerably higher than those measured during prototype tests, even taking into strain rate data may also be needed to assess adhesive performance. account that the strains recorded by each individual loading sensor may be summed in the worst case. Also, Further studies now underway are focusing on developing design criteria for adhesive connections. prototype measurements from the tension-compression

Fig. 1

 1 Fig. 1 (a) Prototype motorboat and (b) detail of the bulkhead-hull

Fig. 2

 2 Fig. 2 Assembly strength versus bondline thickness, single-lap

Fig. 3 Fig. 4

 34 Fig. 3 Mechanical loads on motorboat joints

Fig. 5

 5 Fig. 5 Tension-compression sensor: (a) gauge positions;

Fig. 6 Fig. 7

 67 Fig. 6 Shear sensor

Fig. 9

 9 Fig. 9 Sea trial recordings during changes in direction

  boats. Large savings in assembly time were achieved 6 Huther, M., Roy, A., Gacougnolle, J. L., Goncharova, E.compared with the traditional overlamination method. and Davies, P. Adhesive bonding of steel and fibreglass Drop tests and sea trials, using specially adapted strain plates for shipbuilding. In Proceedings of 54th Annual gauge instrumentation, have indicated low strain levels Assembly of the International Institute of Welding, in the adhesive joint, lower during sea trials than during Ljubljana, July 2001. drop tests. The latter may prove to be useful in qualifying 7 Baur, P., Roy, A., Casari, P., Gacougnolle, J. L., the connections, but further tests are needed to validate Choqueuse, D. and Davies, P. Recherche de la nature des this. No damage was observed after any of the tests. efforts transmis par les liaisons structurales d'un bateau. In Proceedings of Mate ´riaux 2002 Conference, Tours, 2002. This suggests that further optimization of the assembly 8 Koelbel, Jr, J. G. Structural design for high speed craft. design may be possible, and studies are continuing. Prof. Boatbuilder, October/November 2000, 31-46. Fatigue and ageing studies are also under way, as these 9 Baley, C. and Cailler, M. Experimental and numerical will govern long-term performance. behaviour of the structure of a 7.7 metre sailing boat at sea. In Proceedings of 3rd IFREMER Conference on Nautical Construction with Composite Materials, Paris,

Table 1

 1 Prototype boat characteristics instrumented and tested under real conditions. This paper will describe these prototype tests.

	Length	5.75 m

Table 2

 2 Drop test conditions

	Test
	number Test configuration
	1	Boat empty, 1 m drop, horizontal
	2	Boat empty, 2 m drop, horizontal
	3	Boat empty, 1 m drop, on port side in order to increase
		response of tension-compression and bending sensors
	4	Boat empty, 1 m drop, on starboard side
	5	Boat loaded with 600 kg by sandbag, 1 m drop, horizontal

Table 3

 3 Maximum strains recorded during drop tests

		Maximum value	Drop test
	Sensor	of Dl/l (%)	number
	T-C	-0.27	3-5
	B	0.34	4-5
	S	0.06	3

Table 4

 4 Sea trial maximum values recorded as indicative only. In addition, the values provided by the sensors correspond to the surface strains of the joint.

		Maximum value	Maximum
	Sensor	of signal (mV/V )	value of Dl/l (%)	Test	The strain within the joint is quite complex, as can be shown by a simple finite element analysis, and the value
	T-C	-1.2	-0.12	+45°/sea	will strongly depend on the exact geometry. Modelling
	B S	1.1 0.6	0.1 0.015	Facing the waves +45°/sea	of the whole vessel with the detail necessary to study the hot spots, which are to be expected at the stress

3

  Smith, C. S. Design of Marine Structures in Composite Particular emphasis is being placed on rate effects and Materials, 1990 (Elsevier Applied Science, London). the influence of cyclic loading. Finally, it should be noted 4 Roy, A. Comportement me ´canique en sollicitations that drop tests are now being considered by ISO [13] as monotones et cycliques d'assemblages colle ´s compositea means of qualifying small boat structures.

		composite et composite-acier. PhD thesis, Poitiers University,
		1994.
	7 CONCLUSION	5 Roy, A., Gontcharova-Benard, E., Gacougnolle, J. L. and Davies, P. Hygrothermal effects on failure mechanisms
		of composites/steel bonded joints. In Time Dependent and
	This study has demonstrated the strong potential for	Nonlinear Effects in Polymers and Composites, ASTM
	adhesive bonding of bulkhead assemblies in small motor-	Special Technical Publication, STP 1357, 2000, pp. 353-71
		(ASTM ).

ACKNOWLEDGEMENTS 1992, paper 41, pp. 423-431. 10 Hentinen, M. and Holm, G. Load measurements on the

The contributions to this project from L. Frimas of


Grigorios Megariotis

Antonia Vyrkou

Adrien Leygue

Doros N Theodorou

Systematic Coarse Graining of 4-Cyano-4′-pentylbiphenyl

A coarse-grained model is derived for a liquid-crystal-forming molecule, 4-cyano-4′-pentylbiphenyl (5CB), from a detailed atomistic model using the iterative Boltzmann inversion (IBI) method in the isotropic phase at 315 K and 1 bar. The coarse-grained model consists of five "superatoms" (one for the cyano group, two for the aromatic rings in the biphenyl moiety, and two for the alkyl tail), which are categorized as three types. A modification of IBI, wherein only one of the effective intermolecular potentials (the one corresponding to the superatom pair whose intermolecular correlation function exhibits the highest deviation from the atomistic one) is updated at each iteration, proves to be necessary to achieve convergence. The coarse-grained model, which enables a savings of a factor of 35 in computational cost relative to atomistic simulation, is used to explore ordering into liquid-crystalline phases at lower temperatures. It is found to yield a first-order ordering transition at 288 K with small hysteresis and negligible system size effects. A detailed investigation in terms of various structural and dynamical measurements indicates that the ordered phase is of the smectic type rather than nematic, as observed experimentally. The ordering temperature can be brought close to the experimental value of 308.5 K through the simple rescaling of the intermolecular effective interaction potentials employed in the coarse-grained model. A nematic ordered phase can be obtained from the coarse-grained model by scaling up the head-head and tail-tail effective interaction potentials obtained by IBI.

Introduction

Molecular simulations have been employed extensively over the last 2 decades to study soft matter (polymers, liquid crystals, colloids, surfactants, etc.). Despite the rapid increase of computing power available on modern machines and the possibility of massive parallelization, atomistic simulations are currently confined to a few hundreds of nanoseconds in molecular dynamics (MD) or a few hundred million moves in Monte Carlo simulations. To access the long time and length scales that govern the behavior of soft-matter systems, there is a need to change the level of description. This is the idea behind mesoscopic simulation methods, such as coarse-grained MD, Langevin dynamics, Brownian dynamics, lattice Boltzmann, dissipative particle dynamics, and dynamic density functional theory. [START_REF] Zwanzig | Ensemble Method in the Theory of Irreversibility[END_REF][START_REF] Mori | Collective Motion and Brownian Motion[END_REF][START_REF] Carmesin | The Bond Fluctuation Method: a New Effective Algorithm for the Dynamics of Polymers in all Spatial Dimensions[END_REF][START_REF] Groot | Dissipative Particle Dynamics: Bridging the Gap between Atomistic and Mesoscopic Simulation[END_REF][START_REF] Fraaije | The Dynamic Mean-Field Density Functional Method and its Application to the Mesoscopic Dynamics of Quenched Block Copolymer Melts[END_REF] To retain a maximum level of predictive power, it is highly desirable that the models invoked in mesoscopic simulations be based directly on detailed atomistic ones. The derivation of a less detailed model, cast in terms of fewer degrees of freedom, from a detailed atomistic one is termed "coarse graining".

The idea of coarse graining has been widely used in molecular simulations. [START_REF] Nielsen | Coarse Grain Models and the Computer Simulation of Soft Materials[END_REF] In the present work, coarse graining entails the gathering of small units to form larger quasi-spherical entities, with the use of appropriate effective potentials among the newly defined larger entities (also called "beads" or "superatoms"). Even an atomistic model can be regarded as resulting from a coarse-graining procedure; the electrons are incorporated into their nuclei, and each atom is represented by a single sphere. Another familiar example of simple coarse graining is the commonly used united-atom or anisotropic united-atom representation, [START_REF] Martin | Transferable Potentials for Phase Equilibria United-Atom Description of n-Alkanes[END_REF][START_REF] Nath | On the Simulation of Vapor-Liquid Equilibria for Alkanes[END_REF][START_REF] Ungerer | Applications of Molecular Simulation in the Oil and Gas Industry[END_REF] in which hydrogens are absorbed by their vicinal carbons. An important advantage of such an approximation is the elimination of the fastest degrees of freedom, which are vibrational modes associated with the explicit hydrogens. In the same fashion, a mesoscopic model can be constructed from an atomistic one. Properties computed by mesoscopic models, such as elastic constants, viscosities, etc., can, in principle, be used as inputs to continuum models, which employ volume elements to represent a material. [START_REF] Sonnet | Continuum Theory for Nematic Liquid Crystals with Tensorial Order[END_REF][START_REF] Qian | Generalized Hydrodynamic Equations for Nematic Liquid Crysyals[END_REF] A variety of coarse-graining techniques have been presented in the literature. For polymer systems, iterative Boltzmann inversion (IBI) is a popular and rather straightforward method for constructing coarse-grained models. [START_REF] Mu ¨ller-Plathe | Coarse-Graining in Polymer Simulation: From the Atomistic to Mesoscopic Scale and Back[END_REF][START_REF] Reith | Deriving Effective Mesosclale Potentials from Atomistic Simulations[END_REF][START_REF] Sun | Systematic Coarse-Graining of Atomistic Models for Simulation of Polymeric Systems[END_REF][START_REF] Milano | Mapping Atomistic Simulations to Mesoscopic Models: A Systematic Coarse-Graining Procedure for Vinyl Polymer Chains[END_REF][START_REF] Vettorel | Coarse-Graining of Short Polythylene Chains for Studying Polymer Crystallization[END_REF][START_REF] Jain | Do Inverse Monte Carlo Algorithms Yield Thermodynamically Consistent Interaction Potentials[END_REF][START_REF] Spyriouni | Coarse-Grained and Reverse-Mapped United-Atom Simulations of Long-Chain Atactic Polysterene Melts: Structure, Thermodynamic Properties, Chain Conformation, and Entanglements[END_REF][START_REF] Kamio | Coarse-Grained End Bridging Monte Carlo Simulations of Poly(ethyleneterephthalate) Melt[END_REF] Moreover, Mu ¨ller-Plathe and co-workers have developed the automated simplex optimization method. [START_REF] Faller | Automatic Parameterization of Force Fields for Liquids by Simplex Optimization[END_REF][START_REF] Meyer | Coarse Graining of Nonbonded Inter-Particle Potentials Using Automatic Simplex Optimization to Fit Structural Properties[END_REF][START_REF] Reith | Mapping Atomistic to Coarse-Grained Polymer Models Using Automatic Simplex Optimization to Fit Structural Properties[END_REF][START_REF] Faller | Automatic Coarse Graining of Polymers[END_REF] Biomolecules (proteins, lipids, DNA, etc.) constitute another area of significance in which phenomena cannot be captured by atomistic models because of the time and length scales on which they occur; hence, the development of accurate mesoscopic models is necessary. [START_REF] Marrink | Coarse Grained Model for Semiquantitative Lipid Simulations[END_REF][START_REF] Shelley | A Coarse Grain Model for Phospholipid Simulations[END_REF][START_REF] Stevens | Insights into the Molecular Mechanism of Membrane Fusion from Simulation: Evidence for the Association of Splayed Tails[END_REF][START_REF] Tozzini | Coarse-Grained Models for Proteins[END_REF] An example of such a model, developed heuristically to match atomistic simulation results and experiment, is that of Marrink et al. for lipid-bilayer membranes. [START_REF] Marrink | Coarse Grained Model for Semiquantitative Lipid Simulations[END_REF] A promising development in the field of systematic coarse graining is the force-matching method of Voth and collaborators, which has been applied successfully to lipid-water systems. [START_REF] Izvekov | A Multiscale Coarse-Graining Method for Biomolecular Systems[END_REF][START_REF] Izvekov | Multiscale Coarse Graining of Liquid-State Systems[END_REF][START_REF] Noid | Multiscale Coarse-Graining and Structural Correlations: Connections to Liquid-State Theory[END_REF][START_REF] Voth | Coarse-Graining of Condensed Phase and Biomolecular Systems[END_REF] Coarse-grained models have also been developed systematically by the inverse Monte Carlo method. [START_REF] Lyubartsev | Calculation of Effective Interaction Potentials from Radial Distribution Functions: A Reverse Monte Carlo Approach[END_REF] The objective of the work described in this article is systematically to coarse-grain a molecular system capable of forming thermotropic liquid-crystalline (LC) phases. This study is motivated by recent experiments that indicate that LC phases are capable of interrogating the structure of phospholipidic interfaces at the scale of nanometers, [START_REF] Brake | Biomolecular Interactions at Phospholipid-Decorated Surfaces of Liquid Crystals[END_REF][START_REF] Lockwood | Self-Assembly of Surfactants and Phospholipids at Interfaces between Aqueous Phases and Thermotropic Liquid Crystals[END_REF][START_REF] Brake | Formation and Characterization of Phospholipid Monolayers Spontaneously Assembled at Interfaces between Aqueous Phases and Thermotropic Liquid Crystals[END_REF] thereby providing a powerful tool for probing the nanoscale structure and a basis for the development of sensors for biological molecules. [START_REF] Brake | Biomolecular Interactions at Phospholipid-Decorated Surfaces of Liquid Crystals[END_REF] Because the time and length scales involved in these experiments are clearly out of the range of molecular simulations, there is a pressing need for reliable coarse-grained models. Liquid crystals, a class of substances with numerous and versatile applications in science and technology, [START_REF] Chandrasekhar | Liquid Crystals[END_REF][START_REF] Denk-Ke | Fundamentals of Liquid Crystal DeVices[END_REF] have been simulated extensively with coarse-grained models, based primarily on Gay-Berne ellipsoids. [START_REF] Gay | Modification of the Overlap Potential to Mimic a Linear Site-Site Potential[END_REF][START_REF] Frenkel | Computer Simulation of Hard-Core Models for Liquid Crystals[END_REF][START_REF] Allen | Molecular Dynamics Simulation Using Hard Particles[END_REF][START_REF] Whittle | Liquid Crystal Formation in a System of Fused Hard Spheres[END_REF][START_REF] Allen | Hard Convex Body Fluids[END_REF][START_REF] Williamson | Liquid Crystalline Phase Behavior in Systems of Hard-Sphere Chains[END_REF] Fewer simulations of thermotropic liquid crystals can be found in the literature based on detailed atomistic or united-atom models. [START_REF] Wilson | Molecular Modelling of Liquid Crystal Systems: An Internal Coordinate Monte Carlo Approach[END_REF][START_REF] Lansac | Microscopic Structure and Dynamics of a Partial Bilayer Smectic Liquid Crystal[END_REF][START_REF] Zakharov | Structure and Elastic Properties of a Nematic Liquid Crystal: A Theoretical Treatment and Molecular Dynamics Simulation[END_REF][START_REF] Zakharov | Theoretical Investigations of Rotational Phenomena and Dielectric Properties in a Nematic Liquid Crystal[END_REF][START_REF] Cheung | Parameterization and Validation of a Force Field for Liquid-Crystal Forming Molecules[END_REF][START_REF] Cacelli | Atomistic Simulation of a Nematogen Using a Force Field Derived from Quantum Chemical Calculations[END_REF][START_REF] De Gaetani | Modeling a Liquid Crystal Dynamics by Atomistic Simulation with Ab Initio Derived Force Field[END_REF][START_REF] Tiberio | Towards in Silico Liquid Crystals. Realistic Transition Temperatures and Physical Properties for n-Cyano-Biphenyls via Molecular Dynamics Simulations[END_REF] Attempts to develop coarse-grained models for LC systems by the application of a systematic method have been scarce. [START_REF] Peter | Classical Simulations from the Atomistic to the Mesoscale and back: Coarse Graining an Azobenzene Liquid Crystal[END_REF] The particular LC system that we have chosen for our study is 4-cyano-4′-pentylbiphenyl (5CB), an archetypal example of small-molecule liquid crystals with numerous technological applications. [START_REF] Chandrasekhar | Liquid Crystals[END_REF][START_REF] Denk-Ke | Fundamentals of Liquid Crystal DeVices[END_REF] Extensive experimental studies are available for 5CB. A number of atomistic models have been developed for this system and for related systems in the bulk. [START_REF] Wilson | Molecular Modelling of Liquid Crystal Systems: An Internal Coordinate Monte Carlo Approach[END_REF][START_REF] Lansac | Microscopic Structure and Dynamics of a Partial Bilayer Smectic Liquid Crystal[END_REF][START_REF] Zakharov | Structure and Elastic Properties of a Nematic Liquid Crystal: A Theoretical Treatment and Molecular Dynamics Simulation[END_REF][START_REF] Zakharov | Theoretical Investigations of Rotational Phenomena and Dielectric Properties in a Nematic Liquid Crystal[END_REF][START_REF] Cheung | Parameterization and Validation of a Force Field for Liquid-Crystal Forming Molecules[END_REF][START_REF] Cacelli | Atomistic Simulation of a Nematogen Using a Force Field Derived from Quantum Chemical Calculations[END_REF][START_REF] De Gaetani | Modeling a Liquid Crystal Dynamics by Atomistic Simulation with Ab Initio Derived Force Field[END_REF][START_REF] Tiberio | Towards in Silico Liquid Crystals. Realistic Transition Temperatures and Physical Properties for n-Cyano-Biphenyls via Molecular Dynamics Simulations[END_REF] Our aim here is to develop a mesoscopic model for 5CB that will be optimized in the isotropic phase for some chosen structural and thermodynamic properties (radial distribution functions, intramolecular correlations, and density) and then test the ability of this model to capture the ordering transition and the structure and density of the ordered phase. The method we chose to apply for coarse graining is IBI. Specific questions we address are as follows: (a) Is it possible to coarse-grain 5CB in the isotropic phase, starting from a detailed atomistic model, through the IBI procedure? (b) If not, how can the IBI procedure be modified to achieve convergence to the target distribution functions? (c) Does a coarse-grained model so derived using structural information from the isotropic phase capture the first-order transition into the LC phase qualitatively and quantitatively? (d) How can the quantitative performance of the model in the ordered phase be improved? It has to be mentioned that this is the first time that the IBI method, as used for polymer systems, is applied in a LC system (5CB). A related method has been employed for nonbonded interactions in a previous study, [START_REF] Peter | Classical Simulations from the Atomistic to the Mesoscale and back: Coarse Graining an Azobenzene Liquid Crystal[END_REF] but in that case, the effective potentials were computed by simulating phases formed by fragments of the LC molecules.

This paper is organized as follows. The coarse-grained model that we developed is introduced in section 2. In section 3, we provide a brief description of the IBI method, discuss problems that we encountered with it, and introduce a modification that we designed in order to achieve convergence in the case of 5CB molecules. In section 4, we provide details on the simulations that we have conducted. The performance of our coarse-graining strategy in the isotropic phase is discussed in section 5, where convergence of the algorithm is quantified via appropriate figures and tables. Section 6 reports on our coarse-grained simulations of the ordering transition. Section 7 is devoted to a modification of our coarse-grained model aimed at improving the description of the ordered phase and to the structural and thermodynamic predictions that we obtained from the modified model. Section 8 summarizes the main conclusions from this study.

Molecular Models and Measurements of the Structures

All coarse-grained descriptions presented in this work have been developed from the same parent atomistic model, that of ref 51. The atomistic force field employs stretching, bending, and torsional intramolecular contributions along with Coulombic and Lennard-Jones intermolecular interactions. It adopts a united-atom representation for the alkyl tails and aromatic rings of the 5CB molecules. A total of 19 sites are used to represent each 5CB molecule.

In the coarse-grained (mesoscopic) model that we developed, the 19 atomistic sites were replaced by five superatoms. The coarse-grained representation is shown in Figure 1, along with an atomistic one. The first step toward the development of a coarse-grained model is mapping, i.e., defining the coordinates of each superatom on the atomistically represented molecule. At the mesoscopic level, the centers of superatoms CT1 and CT2 (group type CGCT) of the alkyl tail are placed at the positions of its second and fourth atomistic carbons, respectively. The positions of the superatoms AC1 and AC2 (group type CGAC), which represent the aromatic rings, are chosen to coincide with the atomistic centers of the rings, while the position of CNZ (group type CGCNZ) is chosen as the center of the atomistic CtN bond. Table 1 provides the molar masses of the superatoms in the context of the mapping adopted here. Now we turn our attention to the intramolecular potentials of a 5CB molecule at the mesoscopic level. Intramolecular potentials are applied here only for effective bending angles (θ) and distance (r)-dependent intermolecular interactions. Applying our coarse-graining strategy to a fully flexible mesoscopic model led to narrow Gaussian distributions of the effective bond lengths (l; see the Supporting Information). Using fixed effective bond lengths in place of these distributions left all structural and thermodynamic properties obtained at the mesoscopic level unchanged, while at the same time affording an increase in the time step for integration of the equations of motion of the mesoscopic model by more than a factor of 2. Thus, it was decided to use fixed l in the production phase of all coarse-grained simulations. The constant l values employed are given in the caption to Figure 1. As far as effective torsion angles (φ) are concerned, only torsion around the AC1-CT2 effective bond needs to be considered with the coarse-grained model adopted here. The difference between the minimum and maximum in the effective potential derived by Boltzmann inversion of this effective torsion angle, however, was much less than k B T. Thus, no effective torsional potentials were incorporated in the coarse-grained model.

The number of intermolecular effective potentials required at the coarse-grained level is n(n + 1)/2 ) 6, with n ) 3 being the number of superatom types. The development of coarsegrained potentials could be based on nonbonded interactions, instead of intermolecular ones. In this study, however, we preferred to separate intermolecular nonbonded interactions from possible intramolecular nonbonded interactions. The six inter- molecular potentials were developed by the iterative strategy described in sections 3 and 4 and stored as functions of separation r in tabular form.

We have evaluated an alternative coarse-grained model that, in addition to the superatoms and interactions described above, contained two partial charges on virtual atoms representing the carbon and nitrogen of the cyano group. The values of the partial charges reproduced the atomistic dipole moment of the 5CB molecule at the mesoscopic level. Good convergence to the target distribution functions was achieved for this coarse-grained model with explicit charges with the method described in sections 3 and 4. It was observed, however, that this model tended to form "glassy" phases, rather than ordered LC phases, upon cooling. This failure to predict the correct thermodynamic behavior can be explained on the basis of effective intermolecular potentials: The electrostatic interactions of the atomistic model have already been taken into account in deriving the effective potentials of the coarse-grained model (see also section 3); consequently, the use of explicit charges leads to double counting of the interactions in question. The model with explicit charges was therefore abandoned, and the simpler (and less expensive computationally) charge-free coarse-grained model outlined above was adopted for the production phase of our calculations.

Experimentally, 5CB is a uniaxial thermotropic LC molecule that displays only a nematic phase between its isotropic and crystalline phases. There are, however, other CB (cyanobiphenyl) molecules that exhibit both nematic and smectic phases (e.g., 8CB). [START_REF] Luckhurst | The Molecular Physics of Liquid Crystals[END_REF][START_REF] Hird | Physical Properties of Liquid Crystals[END_REF] A central role in our analysis of ordering will be played by the Saupe ordering tensor, [START_REF] De Gennes | The Physics of Liquid Crystals[END_REF] defined by the following equation:

In eq 1, N is the number of molecules, u ˆi is the unit vector along the axis of the ith molecule, and 1 is the unitary tensor of second order. The (dyadic) tensor Q is obtained as an instantaneous value for each configuration. The molecular axis is defined as the long axis obtained by diagonalization of the moment of inertia tensor, 56 I: Equation 2 is applied to each molecule, and the sum is taken over all atoms a that constitute one molecule. The atomic position vectors r a are defined with respect to the molecular center of mass.

As defined, the tensor Q is symmetric and traceless, so it will have three real eigenvalues that sum to zero. In a phase characterized by cylindrical symmetry, Q will have one positive and two equal negative eigenvalues. As a criterion for the transition from the isotropic to the LC phase and vice versa, we will consider the order parameter S, defined as the largest eigenvalue of Q. An alternative, but equivalent, definition of the order parameter equates it with -2 times the middle eigenvalue of the Q tensor. [START_REF] Eppenga | Monte Carlo Study of the Isotropic and Nematic Phases of Infinitely Thin Hard Platelets[END_REF] In our ordered phases, the molecules will be oriented, on average, along a specific direction, which is specified by a unit vector called the director, n. The director is defined here as that eigenvector of Q that corresponds to its largest eigenvalue. Instantaneous order parameters and directors are defined for each configuration and averaged over the production phase of a run.

Practically, values of the order parameter from 0.45 to 0.6 indicate the existence of a nematic phase. [START_REF] Luckhurst | The Molecular Dynamics of Liquid Crystal; NATO ASI Series[END_REF] In the isotropic phase, the order parameter tends to zero, with an increasing number of molecules N as ∼N -1/2 (see section 5).

Coarse-Graining Method

The application of IBI presupposes a description of each molecule at the mesoscopic level as a set of superatoms. The many-body joint distribution function for all superatom coordinates could, in principle, be derived from atomistic simulation, and a potential of mean force could be defined by the logarithm of this joint distribution function. In place of the rigorous manybody potential of mean force, IBI uses a combination of effective potentials, each depending on a particular subset of the superatom degrees of freedom, to approximate it. The IBI procedure identifies this particular approximation as the unique decomposition that reproduces a particular set of lower-order distribution functions. In particular, it has been proven that there is a unique relationship between a pair potential and its corresponding radial distribution function (RDF). [START_REF] Lyubartsev | Calculation of Effective Interaction Potentials from Radial Distribution Functions: A Reverse Monte Carlo Approach[END_REF][START_REF] Henderson | A Uniqueness Theorem for Fluid Pair Correlation Functions[END_REF][START_REF] Lyubartsev | Osmotic and Activity Coefficients from Effective Potentials for Hydrated Ions[END_REF][START_REF] Soper | Empirical Potential Monte Carlo Simulation of Fluid Structure[END_REF] In general, IBI considers contributions from effective bond lengths (l), effective bond angles (θ), and effective torsion angles (φ) formed by the superatoms, as well as distance-dependent effective intermolecular pair potentials between the superatoms. The results presented here were obtained with a model employing constant effective bond lengths and no effective torsional potentials. First estimates for the intramolecular bending potentials and for the intermolecular potentials between superatoms were obtained from the probability densities of the three θ's and the six intermolecular pair distribution functions between superatom types accumulated in the course of an equilibrated detailed atomistic simulation [START_REF] Mu ¨ller-Plathe | Coarse-Graining in Polymer Simulation: From the Atomistic to Mesoscopic Scale and Back[END_REF][START_REF] Reith | Deriving Effective Mesosclale Potentials from Atomistic Simulations[END_REF] (eqs 3 and 4):

The functions P(θ) are atomistically computed intramolecular probability densities for the effective bond angles θ, while the factor sin θ arises in the Jacobian of transformation from Cartesian to generalized coordinates. [START_REF] Gφo | On the Use of Classical Statistical Mechanics in the Treatment of Polymer Chain Conformation[END_REF] Equation 4 contains a RDF, g Rβ (r), for the pair of superatom species (R, β), obtained from the atomistic trajectory as well. In eqs 3 and 4, the prelogarithmic factor contains the temperature T of the system and the Boltzmann constant k B (1.381 × 10 -23 J/K), which emphasizes that U 1 bend (θ) and U Rβ,1 inter (r) are not real potentials but configurational free energies. Additional temperature and density dependence arises from the distributions within the logarithmic factors.

Detailed descriptions of the IBI method are given in many previous simulation studies. [START_REF] Mu ¨ller-Plathe | Coarse-Graining in Polymer Simulation: From the Atomistic to Mesoscopic Scale and Back[END_REF][START_REF] Reith | Deriving Effective Mesosclale Potentials from Atomistic Simulations[END_REF][START_REF] Sun | Systematic Coarse-Graining of Atomistic Models for Simulation of Polymeric Systems[END_REF][START_REF] Milano | Mapping Atomistic Simulations to Mesoscopic Models: A Systematic Coarse-Graining Procedure for Vinyl Polymer Chains[END_REF][START_REF] Vettorel | Coarse-Graining of Short Polythylene Chains for Studying Polymer Crystallization[END_REF][START_REF] Jain | Do Inverse Monte Carlo Algorithms Yield Thermodynamically Consistent Interaction Potentials[END_REF][START_REF] Spyriouni | Coarse-Grained and Reverse-Mapped United-Atom Simulations of Long-Chain Atactic Polysterene Melts: Structure, Thermodynamic Properties, Chain Conformation, and Entanglements[END_REF][START_REF] Kamio | Coarse-Grained End Bridging Monte Carlo Simulations of Poly(ethyleneterephthalate) Melt[END_REF] In our work, we found that the standard IBI method, when applied in the isotropic phase of 5CB in conjunction with the atomistic model discussed above, failed to converge (see Figure 2). This necessitated the development of a modified IBI method. Instead of globally updating all intermolecular effective potentials U Rβ inter at every iteration, our modified procedure updates only one U Rβ inter at a time. Specifically, each time we choose to modify that intermolecular effective potential that corresponds to the RDF of maximum error relative to the atomistically computed target RDF. A flow diagram of our modified IBI method is given in Scheme 1.

To correct the density or pressure of the system (depending on what ensemble is simulated), a ramp correction (or else "linear correction") was applied:

Q ) 1 N ∑ i)1 N { 3 2 u ˆiu ˆi - 1 2 1 } (1)
I ) ∑ a m a (r a 2 1 -r a r a ) (2)
U 1 bend (θ) ) -k B T ln[P(θ)/sin θ] (3) U Rβ,1 inter (r) ) -k B T ln[g Rβ (r)] (4)
The left-hand side of the previous equation contains the corrected intermolecular potentials U Rβ corr (r) between the superatoms R and β. The second term on the right-hand side of eq 5 depends on the temperature and on the cutoff distance r c of the intermolecular potentials, and C is an adjustable dimensionless parameter used to modify the pressure P. C is here taken to be the same for all pairs of superatom types. An estimate of C can be obtained through the pressure equation with N mol , N R , and V standing for the total number of molecules, the total number of superatoms of type R, and the volume of the system, respectively.

Substituting eq 5 in eq 6, the following equation in C is obtained:

∆P represents the difference between the target (atomistic) value of pressure and the value derived from a coarse-grained simulation using IBI effective potentials. The need to adjust the pressure after implementation of IBI is attributed to the fact that the virial pressure is not retained upon coarse graining, even if the atomistically derived g Rβ (r) functions are reproduced. [START_REF] Izvekov | Multiscale Coarse Graining of Liquid-State Systems[END_REF] In eqs 5 and 7, r c and the adjustable dimensionless parameter C are defined to be the same for all superatom pairs (R, β).

Simulation Details

All simulations were performed with the MD package [START_REF] Berendsen | GROMACS: A Message-Passing Parallel Molecular Dynamics Implementation[END_REF][START_REF] Van Der Spoel | GROMACS: Fast, Flexible and Free[END_REF][START_REF] Van Der Spoel | GROMACS User Manual, version 3.3[END_REF] GROMACS 3.3.2. Effective bond lengths in the coarse-grained model were constrained to their average values using the SHAKE 66 algorithm (relative tolerance 10 -5 ). This allowed the use of a time step of 20 fs. By comparison, the time step used in connection with the atomistic model was 2 fs. [START_REF] Tiberio | Towards in Silico Liquid Crystals. Realistic Transition Temperatures and Physical Properties for n-Cyano-Biphenyls via Molecular Dynamics Simulations[END_REF] The atomistic trajectories employed in this work were produced by repeating all simulations in the GROMACS package with the same simulation details as those reported in ref 51 except for the time step, which was equal to 2 fs in our simulations. All structural and thermodynamic information derived from our atomistic simulations was within error bars of those reported in ref 51.

The coarse-grained MD simulations involved in the IBI iterations were conducted in the NVT and NPT ensembles using the Nose ´-Hoover thermostat [START_REF] Nose | A Molecular Dynamics Method for Simulations in the Canonical Ensemble[END_REF][START_REF] Hoover | Canonical Dynamics: Equilibrium Phase-Space Distributions[END_REF] with a time constant equal to 1.0 ps. In the NPT ensemble, the simulation box was kept rectangular and the pressure was set to P ) 1 bar using the GROMACS [START_REF] Van Der Spoel | GROMACS User Manual, version 3.3[END_REF] version of the Parrinello-Rahman barostat [START_REF] Parrinello | Polymorphic Transitions in Single Crystals: A New Molecular Dynamics Method[END_REF][START_REF] Nose | Constant Pressure Molecular Dynamics for Molecular Systems[END_REF] with a time constant equal to 4.0 ps. In order to obtain accurate structural properties, free of statistical noise, every IBI iteration involved a coarse-grained MD simulation of one million steps.

For the intermolecular interactions of the coarse-grained model, a simple cutoff scheme was employed with a r c distance equal to 1.5 nm, where intermolecular potentials tend to zero or, equivalently, the RDFs between superatoms tend to unity. These interactions, as obtained by the IBI method, were stored in tabular form, and a smoothing via Chebyshev polynomials [START_REF] Galassi | GNU Scientific Library Reference Manual[END_REF][START_REF] Broucke | Algorithm: Ten Subroutines for the Manipulation of Chebyshev Series[END_REF]

U Rβ corr (r) ) U Rβ (r) -Ck B T ( 1 -r r c ) (5)
PV ) N mol k B T - 1 6V ∑ R ∑ β N R N β ∫ 0 +∞ r dU Rβ (r) dr g Rβ (r) 4πr 2 dr (6) V∆P ) - 2π 3
Ck B T Vr c ∑ R ∑ β N R N β ∫ 0 r c g Rβ (r) r 3 dr (7)
was applied (full details are given in the Supporting Information). The effectiveness of this method is depicted in Figure 3, where a noisy effective potential is approximated by a smooth function. Intramolecular interactions were fitted by functional forms available in GROMACS 3.3.2 (see also the Supporting Information for effective bending potentials). In all cases, the fitting involved a least-squares approximation of the numerical potentials.

The excluded volume region, readily detected in the RDFs between superatoms, was described by a simple repulsive potential of an inverse 12th degree (A/r 12 ). The parameter A was computed under the demand of continuity between this extrapolated repulsive potential and the IBI potentials at the connection point. The best value of parameter C in the ramp correction of eq 5 was found to be equal to 1.5 × 10 -4 .

Because of the elimination of some degrees of freedom for 5CB, the dynamics appears faster in comparison to that of the atomistic model. In order to obtain similar self-diffusivities of 5CB between the two levels of description, a friction force is introduced in the context of Langevin dynamics. The friction constant value that proved to work well in this work equals 4.5 ps -1 . Details on how stochastic dynamics is applied are given in ref 65. All other parameters are exactly the same as those of the MD simulations, reported above.

The IBI method was initially applied to a coarse-grained system of 216 5CB molecules in the isotropic phase (315 K and 1 bar). In addition, in order to check for possible system size effects on structural properties, larger isotropic systems of 5CB, consisting of 432, 864, and 1728 molecules, were simulated at the coarse-grained level. Initial configurations for these systems were constructed by appropriate transformations of the primary simulation box of 216 molecules. Neither distribution functions nor density exhibited any dependence on the size of the simulated system.

After the development of the coarse-grained model in the isotropic phase, the structure of the system was investigated in the LC region. To this end, MD simulations were conducted with the temperature changing linearly with time at a prescribed rate of cooling or heating (annealing mode of GROMACS). [START_REF] Van Der Spoel | GROMACS User Manual, version 3.3[END_REF] All other conditions of the ordering coarse-grained simulations were exactly the same as those reported above for simulations of the isotropic phase.

Performance of the Coarse-Graining Algorithm in the Isotropic Phase

In this section, we first discuss the ability of the developed coarse-grained model to reproduce the chosen atomistic structural properties in the isotropic phase of 5CB at 315 K and 1 bar. The crucial part for developing this model was the convergence of RDFs. For this reason, the iterative coarsegraining procedure was monitored via relative errors defined by the following equation:

The RDFs of the ith iteration and the atomistic target function for superatom pair (R, β) are symbolized byg Rβ,i (r) and g Rβ (r), respectively. The weighting function w Rβ (r) in these integrals may be used to give more weight to some points of the grid. In our case, this function was set equal to exp(-r/r*) with r* ) 1 nm. Table 2 lists the relative errors for all RDFs for the first, second, and final iterations of the IBI method. It is concluded from the third row of the aforementioned table that a high quality of convergence was achieved. The success of the modified IBI method is also shown clearly in Figure 4, where three of the six RDFs are presented (see the Supporting Information for the other three). In each diagram, three lines are depicted, displaying the g Rβ (r) obtained from the atomistic simulation, the g Rβ,1 (r) from the first iteration of the IBI algorithm, and the final converged distribution function g Rβ,15 (r) describing the structure at the mesoscopic level. A reminder of how the IBI algorithm is started would perhaps be useful here. First, the RDFs and intramolecular distributions are computed by the atomistic model for the given mapping. Second, the aforementioned quantities are substituted in eqs 3 and 4. With this procedure, the first set of intra-and intermolecular potentials are available so as to start the first simulation (i.e., first iteration of IBI) at the mesoscopic level.

As far as the density of the system at the mesoscopic level is concerned, there is very good agreement with both the parent atomistic model [START_REF] Tiberio | Towards in Silico Liquid Crystals. Realistic Transition Temperatures and Physical Properties for n-Cyano-Biphenyls via Molecular Dynamics Simulations[END_REF] and experimental measurements. [START_REF] Wuerflinger | Physical Properties of Liquid Crystals[END_REF] Although the optimization procedure took place at a specific temperature (315 K), we present in Figure 5 the results for three different temperatures in the isotropic phase. In addition, the densities are compared with those reported for another atomistic model. [START_REF] Cacelli | Atomistic Simulation of a Nematogen Using a Force Field Derived from Quantum Chemical Calculations[END_REF] It is instructive to examine system size effects on the order parameter S of the isotropic phase. To this end, we have simulated systems of 216, 432, 864, and 1728 5CB molecules at the coarse-grained level at 315 K and 1 bar. S was obtained by diagonalizing the instantaneous Q tensor in each recorded configuration and then averaging over all configurations in the equilibrated portion of a run. The statistical error on S thus obtained was also estimated by block averaging [START_REF] Allen | Computer Simulation of Liquids[END_REF] over the equilibrated portion of the run. The results are presented in Figure 6. They can be fitted to an excellent approximation with an expression of the form S(N) ) constant/N 1/2 . In the isotropic phase, S, which is nonnegative by definition (compare the discussion in section 1), behaves as a fluctuation quantity converging to its thermodynamic limit of 0 as N -1/2 .

Closing this section, it has to be mentioned that the diffusion constant of 5CB is not retained upon coarse graining. In particular, it is greater in the case of the coarse-grained model (2.9 × 10 -5 cm 2 /s) in comparison with the corresponding atomistic value (5.5 × 10 -6 cm 2 /s). This problem can be alleviated by applying stochastic dynamics instead of MD. Now, a friction and a random force are introduced so that the dissipation-fluctuation theorem is satisfied. [START_REF] Van Kampen | Stochastic Processes in Physics and Chemistry[END_REF] The value of the friction constant used in our coarse-grained stochastic dynamics simulations was reported in section 4; for this value, the diffusion constant equals 5.3 × 10 -6 cm 2 /s. At this point, it has to be stressed that the correction of the self-diffusion constant was based on a trial-and-error procedure rather than on a

∫ 0 r c w Rβ (r) [g Rβ,i (r) -g Rβ (r)] 2 dr ∫ 0 r c w Rβ (r) [g Rβ (r)] 2 dr ( 8
)
systematic technique for dynamical coarse graining, such as the projection operator formalism. [START_REF] Zwanzig | Ensemble Method in the Theory of Irreversibility[END_REF][START_REF] Mori | Collective Motion and Brownian Motion[END_REF][START_REF] Zwanzig | Memory Effects in Irreversible Thermodynamics[END_REF][START_REF] Mori | A Continued-Fraction Representation of the Time-Correlation Functions[END_REF] An interesting discussion of how this more rigorous formalism can be applied in computer simulations can be found in ref 78.

Coarse-Grained Simulations of the Ordering Transition

We now turn our attention to the ability of the coarse-grained model, as developed by our modified IBI method in the isotropic phase, to predict characteristics of the ordering transition and of the LC phase of 5CB.

Isobaric cooling MD simulations of our IBI-based model, started in the isotropic phase, led readily to ordering, detectable by discontinuous increases in the order parameter S (large) and the density (small). A schematic representation of the isotropic and LC phases at the mesoscopic level is provided in Figure 7.

In this figure, each 5CB molecule is represented by a vector connecting the superatoms AC1 and CNZ. The ordering is visually obvious.

To estimate the first-order transition points from the isotropic to LC phase and vice versa, we conducted coarse-grained MD at decreasing and increasing temperatures. Initially, an isobaric MD simulation was undertaken with a cooling rate of 0.3 K/ns starting from the isotropic phase at 300 K and 1 bar and ending at 278 K. An isobaric heating simulation back to 300 K followed at the same heating rate, 0.3 K/ns. Results from the latter cooling/heating computer experiment are shown in Figure 8. It can be concluded from this figure that the transition points of the IBI-based coarse-grained model from the isotropic to LC phase and vice versa are 281 and 295 K, respectively. As the number of 5CB molecules in the model system increases, the cooling and heating rates for the temperature annealing be lowered to maintain the width of the hysteretic region. Such a hysteresis in the ordering phase transition is not seen experimentally; nevertheless, it is not too large, given the high rates of cooling and heating (compared to the laboratory rates) employed in the simulations. We can estimate the equilibrium temperature for the coexistence of disordered and LC phases of our coarse-grained model as an average from the estimates obtained from cooling and heating simulations, i.e., 288 K. The hysteresis region can be made narrower by reducing the cooling/ heating rate. To explore this, we undertook constant temperature simulations in this region. An initial isotropic phase was found to change into a smectic phase after 720 ns of constanttemperature MD at 285 K, while a smectic phase was found to disorder into an isotropic one after 40 ns of constant-temperature simulation at 292 K. The atomistic model that served as a basis for our coarse-grained model 51 yields a nematic-to-isotropic transition temperature very close to the experimental value of 308.5 K. We conclude that our coarse-grained model with parameters developed to match the atomistic structure in the phase at 315 K underestimates the transition temperature by 20 K. This observation underlines the difficulty of capturing the ordering transition correctly with IBI-based models. Another interesting observation is that the transition is indeed of first order from a thermodynamics point of view. The reader may have noticed our use of the term "liquid-crystalline phase", rather than "nematic phase", for the phase obtained by cooling. This is intentional because the structure of this ordered phase will have to be investigated. Moreover, some important properties of the LC phase are compared and discussed with those obtained from a modified approach in the next section. At this point, one can speculate about the inability of the original model to predict the nematic phase. CB liquid crystals can produce different LC phases according to the length of their alkyl tail. For instance, 5CB forms a nematic phase, while 8CB can be ordered both in nematic and smectic phases. In our case, because of the use of a simple and coarse-grained model for 5CB, the flexibility of the alkyl tail is not reproduced perfectly, and consequently the nematic phase is not formed.

Improved Coarse-Grained Models and the LC

Structure Obtained from Them 7a. Approach to the Smectic Phase. From the above analysis, an important question arises: Can our IBI-based model be modified so as to give a transition temperature close to the experimental value for 5CB? Such a modification (modified model I) was attempted by rescaling the intermolecular interaction potentials (eq 9). In particular, only the intermolecular potentials were changed according to the following scheme:

The prefactor a, common to all coarse-grained intermolecular potentials, was estimated through a trial-and-error procedure, until the predicted transition temperature was within 6 K of the experimental value T NI EXP (308.4 K). Again, all intermolecular potentials U modified (r) were determined in tabular form. In particular, a MD simulation with increasing temperature was undertaken (heating rate equal to 0.3 K/ns), starting from 275 K. The pressure was kept at 1 bar in all three directions. When the prefactor R is equal to 1.05, a very satisfactory transition point (302 K) is obtained. In addition, with the above set of modified potentials, a new ramp correction was applied (C ) -1.0 × 10 -5 ) so as to adjust the density.

Having defined the parameter a, we have a set of intermolecular potentials available in tabular form that allow a study of the modified model in the LC region at higher temperatures than those where a LC phase is observed with the IBI-based model. All simulation results for the LC phase, in the context of the modified model, were collected at 300 K, where the system of coarse-grained 5CB molecules is found in the ordered phase. As regards the orientation of the director, this stays practically unchanged during simulations, as indicated by the normalized autocorrelation function C(t) ) 〈n(0) • n(t)〉 [C(t) > 0.994 for the longest simulation time accessed], depicted in Figure 9. In addition, the same quantity is presented for the IBI-based model at 275 K for a modified model, designed to give a nematic phase at 300 K (for more details, see the next subsection). U modified (r) ) aU(r) (9) A detailed analysis of some important properties was undertaken to characterize the structure of the LC phase. These properties were computed for both the original IBI-based (section 6) and modified coarse-grained model I (present section) at two different temperatures (the modified model at 300 K and the IBI-based model at 275 K). We started with the self-diffusion behavior of LC molecules in the coarse-grained system. In view of the strong anisotropy, separate self-diffusion coefficients were computed along the director and in the plane perpendicular to the director. An important quantity, presented in Table 3, is the ratio of these two diffusion coefficients, D | /D ⊥ . It is seen that D | < D ⊥ for both coarse-grained models, which is evidence for the existence of a smectic phase. [START_REF] Kruger | Diffusion in Thermotropic Liquid Crystals[END_REF] In nematic phases, the following inequality holds: D | > D ⊥ . Thus, the self-diffusion behavior, along with the layered structure observed visually in our ordered phases (see Figure 7b), gives us a first indication for a smectic phase. Some quantitative measurements of the structure are needed as well, however.

The first structural property presented is the probability density of the angle θ between the molecular axis and the director, divided by the factor sin(θ)/2 (see Figure 10). The latter factor is the probability density expected of an isotropic phase, where molecular orientations are random. Three main conclusions are derived from Figure 10: First, there is a strong tendency for molecules to orient parallel or antiparallel to the director. Second, parallel and antiparallel orientations are equally probable. Third, the existence of a smectic C phase is excluded; in such a smectic phase, the director does not coincide with the normal to the layers. The reduced distribution of orientation angles is plotted in Figure 10 at two different temperatures (275 and 300 K for the IBI-based model and its modification, respectively), but corresponding diagrams at different temperatures in the smectic phase are very similar.

Another important structural property is the distribution of molecular centers along the director, depicted in Figure 11. From the form of this distribution, it is clear that we have a layered structure. This further supports the evidence presented above for a smectic phase. In the same figure, we have plotted the distributions that we obtained by simulating systems of 432 and 1728 molecules with the IBI-based model. The distributions are very similar, confirming the absence of system size effects. The difference between these distributions and the distribution obtained for the IBI-based model at 275 K is insignificant. One can discern two characteristic distances in the distributions of Figure 11. This tells us that the smectic phase formed by the coarse-grained model actually consists of bilayers made up of antiparallel 5CB molecules. Such a structure consisting of bilayers of antiparallel molecules is observed in the smectic phase of 8CB. [START_REF] De Gaetani | Computational Study through Atomistic Potentials of a Partial Bilayer Liquid Crystal: Structure and Dynamics[END_REF] Unfortunately, the smectic structure predicted by the coarse-grained model at 1 atm and 300 K is not in agreement with experiments and simulations with the parent atomistic model [START_REF] Tiberio | Towards in Silico Liquid Crystals. Realistic Transition Temperatures and Physical Properties for n-Cyano-Biphenyls via Molecular Dynamics Simulations[END_REF] of 5CB, which give a nematic phase under these conditions.

To confirm that the smectic phase is actually the equilibrium phase of our coarse-grained model at low temperature, we generated a nematic configuration of 216 5CB molecules at 300 K and 1 bar by atomistic simulation; replicated it periodically to generate a quadruple system; replaced the atomistically represented molecules in the resulting system by coarse-grained molecules; and proceeded to simulate the coarse-grained system with isothermal-isobaric MD under the same conditions. The coarse-grained model system readily converted from the nematic to a smectic phase, characterized by the structural features discussed above. 7b. Approach to the Nematic Phase. In the previous subsection, we provided a methodology that allows us to correct the transition temperature with good consistency between the original and modified approaches. However, the ordered structure predicted for 5CB by our coarse-grained model under ambient conditions is incorrect, as pointed out in the previous section. For this reason, we designed a heuristic approach, based on a rescaling of the potentials as described by eq 9, to develop an alternative coarse-grained model that will give a nematic phase at 300 K and 1 atm. Our choice is to rescale only the potentials CGCT-CGCT (tail-tail) and CGCNZ-CGCNZ (head-head) with the parameter R equal to 1.2. For the remaining potentials, a ramp correction (C ) -8 × 10 -6 ) was applied to adjust the density of the system.

Order parameter and density are two crucial parameters that can be compared directly against the parent atomistic model. The comparison in question is recorded in Table 4. Moreover, some additional properties were checked in order to judge the quality of the nematic phase. First, the stability of this phase is explicitly depicted in Figure 9, in which the normalized autocorrelation function of the director is recorded. The smooth appearance of this function in relation to those shown in the same figure for the smectic phase is ascribed to the fact that results from a much larger system (1728 molecules) are depicted in the case of the nematic phase. All ordered phases examined in Figure 9 are clearly very stable because the aforementioned autocorrelation functions of the director remain close to unity. Another criterion that we have a nematic phase is the ratio of the diffusion constants along the director and normal to the director. In Table 3, it is seen that this quantity D | /D ⊥ is greater than unity for the nematic phase that we constructed. Futhermore, the phase does not have a layered structure, as is seen in Figure 11.

Conclusions

A coarse-grained model of 5CB molecules has been developed in this study based on simulations of a detailed atomistic model 51 at 1 bar and 315 K, where 5CB is found in the isotropic phase. The IBI method had to be modified to achieve convergence in our case (liquid of small, strongly polar molecules consisting of three types of coarse-grained sites). Important elements of the modified IBI strategy were (a) a conservative updating scheme, which updates only one (the poorest) effective intermolecular potential at a time, instead of the common global update and (b) the use of Chebyshev polynomials and of a r -12 excluded volume repulsion at short distances to fit the pretabulated numerical effective intermolecular interaction potentials and eliminate the adverse effects of noise in these potentials. The algorithm that we have designed, built around the MD engine of GROMACS 3.3.2, is rather fast, requiring 15 iterations to reproduce the structure and density to within 0.2% and 1.5% of the atomistic simulation, respectively.

In general, the main aim of a coarse-grained model is to augment the time and length scales that can be addressed by simulation. The coarse-grained model that we have developed can be used with a time step of 20 fs, instead of 2 fs required at the atomistic level. Moreover, the degrees of freedom for a 5CB molecule have been reduced drastically to the coordinates of 5 superatoms subject to 4 holonomic effective bond length constraints, in place of 19 atoms in the original united-atom representation. Using this coarse-grained model, our simulations were readily extended up to a simulation box 8 times bigger than the atomistic box of 216 molecules for times of 1 µs. Overall, the coarse-grained model is roughly 35 times more efficient computationally on a single processor than the atomistic model from which it was derived. Of course, simulations of even bigger systems can easily be achieved via parallelization. In addition, we point out that the mesoscopic models presented here do not require electrostatic interactions. Consequently, given the nonideal scaling characterizing commonly used algorithms for electrostatics, they offer an additional advantage for parallelization in comparison with the atomistic model.

We have performed MD simulations of our coarse-grained model, developed in the isotropic phase at 315 K and 1 bar, in order to investigate its ability to predict a transition to a LC phase and the structure of that phase. Our IBI-based model was found to underestimate the transition temperature from the isotropic to oriented phase and vice versa by roughly 20 K. To remedy this, two modified models were developed by heuristic rescaling of the effective intermolecular interaction potentials of the IBI-based model. The first modified model still gives a smectic phase but predicts a transition point of 302 K with relatively small hysteresis (6 K), in good agreement with experiment (308.5 K). On the basis of the order parameter, the predicted transition of this model is clearly first order from a thermodynamic point of view. A number of structural and dynamical characteristics of the LC phase were computed, including the distribution of angles formed with the director, the distribution of molecular centers of mass along the director, and self-diffusivities along and normal to the director. All of these measurements indicate that the ordered structure in the LC phase is smectic. Real 5CB, however, as well as the parent atomistic model, undergoes a transition from the isotropic to nematic phase. For this reason, through a heuristic modification of two effective pair potentials, we developed another modified model II, which predicts a nematic phase around 300 K. The order parameter and density for this model are very close to the atomistic model and the experimental predictions.

Longer CB molecules (e.g., 8CB) form both nematic and smectic phases in their LC region. In view of this, our modified models may be appropriately extended to simulate such liquid crystals. In addition, we intend to simulate the interface water-5CB in the mesoscopic model (300 K) by using modified model II, which gives the correct phase from a thermodynamic point of view, in conjunction with a coarse-grained water model already available in the literature. Our work so far indicates that Marrink's coarse-grained water model is satisfactory in this respect. [START_REF] Marrink | Coarse Grained Model for Semiquantitative Lipid Simulations[END_REF] After coupling the two coarse-grained models, we apply an optimization technique that minimizes deviations from the structural features of the interface computed by the atomistic model, to determine coarse-grained parameters for the water-5CB interactions. Also, a systematic procedure of deriving a nematic phase (without modifications in the intermolecular potentials) is being investigated.

Figure 1 .

 1 Figure 1. Detailed (19-site) and coarse-grained (five-bead) representations of a 5CB molecule. The lengths of the effective bonds CT1-CT2, CT2-AC1, AC1-AC2, and AC2-CNZ are 0.249, 0.380, 0.422, and 0.338 nm, respectively.

Figure 3 .

 3 Figure 3. (a) Example of a fitting (for the CGAC-CGCNZ potential) by Chebyshev polynomials. (b) More detailed snapshot of the fitting.

Figure 2 .

 2 Figure 2. Divergence of the CGCT-CGCT RDF when all potentials are updated simultaneously at each IBI iteration. Scheme 1. Flow Diagram for the Modified IBI Method

Figure 4 .

 4 Figure 4. RDFs of the pairs (a) CGCT-CGCT, (b) CGCT-CGCNZ, and (c) CGCNZ-CGCNZ for the first and final iterations of the modified IBI method and a comparison with the target atomistic RDF.

Figure 5 .

 5 Figure 5. Comparison of orthobaric densities in the isotropic phase, as obtained from NPT simulations using the IBI-based model (Sim), with other (atomistic) simulation results (At1, parent atomistic model; 51 At2, atomistic model of ref 49) and with experiment 81 (Exp).

Figure 6 .

 6 Figure 6. Order parameter S as a function of the number N of 5CB molecules in the model system and a least-squares approximation of the same order parameter for the IBI-based model in the isotropic phase at 315 K and 1 bar.

Figure 7 .

 7 Figure 7. Total of 1728 5CB molecules, represented by vectors connecting beads AC1 and CNZ: (a) in the isotropic phase at 315 K and 1 bar; (b) in the LC phase at 275 K and 1 bar (IBI-based model, box dimensions in nanometers).

Figure 8 .

 8 Figure 8. Evolution of the order parameter with the temperature during cooling and heating runs at 1 bar on the IBI-based model from 300 to 278 K and vice versa. The cooling and heating rates were 0.3 K/ns.

Figure 9 .

 9 Figure 9. Comparison of the normalized time autocorrelation function of the director in the smectic phases (IBI-based model at 275 K and modified model at 300 K) and in the nematic phase at 300 K. Table 3. Diffusion Constants along the Director and in a Plane Normal to the Director for the Coarse-Grained Models under Study a T (K) D | /D ⊥ 275 0.049 ( 0.027 300 0.22 ( 0.06 300 1.149 ( 0.450 a The first row refers to the IBI-based model at 275 K and the second and third rows to the modified models in smectic and nematic phases, respectively, at 300 K.

Figure 10 .

 10 Figure 10. Comparison of probability densities of angle θ between the molecular axis and director divided by the factor (sin θ)/2 for the two coarsegrained models (the IBI-based model at 275 K and the modified model I at 300 K).

Figure 11 .

 11 Figure 11. Comparison of the distributions of 5CB molecular centers along the director for three models (the IBI-based model at 275 K and modified models I and II at 300 K). At 275 K, possible size effects were checked.

Table 1 . Molar Masses of the Beads in a 5CB Molecule at the Mesoscopic Level

 1

	beads	molar mass (g/mol)	beads	molar mass (g/mol)
	CT1	35.5	AC2	76.0
	CT2	35.5	CNZ	26.0
	AC1	76.0

Table 2 . Relative Errors (%) (eq 8) of the RDFs for Three Different Iterations

 2 Relative error of the first iteration in the NVT ensemble. b Relative error of the second iteration in the NVT ensemble. c Relative error of the 15th iteration in NPT ensemble.

		CGCT-CGCT	CGCT-CGAC	CGCT-CGCNZ	CGAC-CGAC	CGAC-CGCNZ	CGCNZ-CGCNZ
	RE 1 a RE 2 b RE 15 c	2.56 1.08 0.111	0.71 0.340 0.0862	1.93 0.861 0.0584	1.00 1.40 0.153	0.294 1.76 0.100	2.08 0.751 0.104

a RE )

Acknowledgment

Financial support from the European Commission via the EU-NSF Specific Targeted Research Project "Modelling Nanostructured Interfaces for Biological Sensors" (MNIBS; Project NMP3-CT-2005-016375) is gratefully acknowledged.

Supporting Information Available: Details of the coarsegraining procedure and additional results from our coarse-