Self-Assembling Systems E-Book
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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
Provides comprehensive knowledge on concepts, theoretical methods and state-of-the-art computational techniques for the simulation of self-assembling systems
- Looks at the field of self-assembly from a theoretical perspective
- Highlights the importance of theoretical studies and tailored computer simulations to support the design of new self-assembling materials with useful properties
- Divided into three parts covering the basic principles of self-assembly, methodology, and emerging topics
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Seitenzahl: 751
Veröffentlichungsjahr: 2016
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Table of Contents
Cover
Title Page
Copyright
List of Contributors
Preface
Chapter 1: Theoretical Studies and Tailored Computer Simulations in Self-Assembling Systems: A General Aspect
1.1 Introduction
1.2 Emerging Self-Assembling Principles
Acknowledgments
References
Chapter 2: Developing Hybrid Modeling Methods to Simulate Self-Assembly in Polymer Nanocomposites
2.1 Introduction
2.2 Methodology
2.3 Results and Discussions
2.4 Conclusions
Acknowledgments
References
Chapter 3: Theory and Simulation Studies of Self-Assembly of Helical Particles
3.1 Introduction: Why Hard Helices?
3.2 Liquid Crystal Phases
3.3 Hard Helices: A Minimal Model
3.4 Numerical Simulations
3.5 Onsager (Density Functional) Theory
3.6 Onsager-Like Theory for the Cholesteric and Screw-Nematic Phases
3.7 Order Parameters and Correlation Functions
3.8 The Physical Origin of Cholesteric and Screw-Like Order
3.9 The Phase Diagram of Hard Helices
3.10 Helical (Bio)Polymers and Colloidal Particles
3.11 Conclusions and Perspectives
Acknowledgments
References
Chapter 4: Self-Consistent Field Theory of Self-Assembling Multiblock Copolymers
4.1 Introduction
4.2 Theoretical Framework: Self-Consistent Field Theory of Block Copolymers
4.3 Numerical Methods of SCFT
4.4 Application of SCFT to Multiblock Copolymers
4.5 Conclusions and Discussions
Acknowledgments
References
Chapter 5: Simulation Models of Soft Janus and Patchy Particles
5.1 Introduction
5.2 Soft Janus Particle Models
5.3 Soft Patchy Particle Models
5.4 Physical Meanings of the Simulation Parameters in Our Models
5.5 GPU Acceleration
5.6 Self-Assembly of Soft Janus and Patchy Particles
5.7 Conclusions
Acknowledgments
References
Chapter 6: Molecular Models for Hepatitis B Virus Capsid Formation, Maturation, and Envelopment
6.1 Introduction
6.2 Molecular Thermodynamics of Capsid Formation
6.3 Electrostatics of Genome Packaging
6.4 Dynamic Structure of HBV Nucleocapsids
6.5 Capsid Envelopment with Surface Proteins
6.6 Summary and Outlook
Acknowledgments
References
Chapter 7: Simulation Studies of Metal–Ligand Self-Assembly
7.1 Introduction
7.2 Modeling Metal–Ligand Self-Assembly
7.3 Self-Assembly of Supramolecular Coordination Complex
7.4 Self-Assembly of Metal–Organic Frameworks [47]
7.5 Conclusion and Outlook
Acknowledgments
References
Chapter 8: Simulations of Cell Uptake of Nanoparticles: Membrane-Mediated Interaction, Internalization Pathways, and Cooperative Effect
8.1 Introduction
8.2
N
-Varied DPD Technique
8.3 The Interaction between NP and Membrane
8.4 Cooperative Effect between NPs during Internalization
8.5 Conclusions
References
Chapter 9: Theories for Polymer Melts Consisting of Rod–Coil Polymers
9.1 Introduction
9.2 Theoretical Models
9.3 Concluding Remarks
References
Chapter 10: Theoretical and Simulation Studies of Hierarchical Nanostructures Self-Assembled from Soft Matter Systems
10.1 Introduction
10.2 Computational Modeling and Methods
10.3 Hierarchical Nanostructures of Block Copolymer Melts
10.4 Hierarchical Aggregates of Block Copolymer Solutions
10.5 Hierarchically Ordered Nanocomposites Self-Assembled from Organic–Inorganic Systems
10.6 Conclusions and Perspectives
Acknowledgments
References
Chapter 11: Nucleation in Colloidal Systems: Theory and Simulation
11.1 Introduction
11.2 Theory of Nucleation
11.3 Order Parameter
11.4 Simulation Methods for Studying Nucleation
11.5 Crystal Nucleation of Hard Spheres: Debates and Progress
11.6 Two-Step Nucleation in Systems of Attractive Colloids
11.7 Nucleation of Anisotropic Colloids
11.8 Crystal Nucleation in Binary Mixtures
11.9 Concluding Remarks and Future Directions
Acknowledgments
References
Chapter 12: Atomistic and Coarse-Grained Simulation of Liquid Crystals
12.1 Introduction
12.2 Thermotropic Liquid Crystal
12.3 Discotic Liquid Crystals
12.4 Chromonic Liquid Crystals
12.5 Conclusion and Outlook
12.6 Acknowledgment
References
Index
End User License Agreement
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Guide
Cover
Table of Contents
Preface
Begin Reading
List of Illustrations
Chapter 1: Theoretical Studies and Tailored Computer Simulations in Self-Assembling Systems: A General Aspect
Figure 1.1 (a) The coordination number in the fluid phase, , is correlated to the isoperimetric quotient of the polyhedron. Here, is a scalar parameter for the sphericity of the shape and coordination number is a measure of the degree of local order. Data points are drawn as small polyhedra, which are grouped according to the assemblies they form. (b) Polyhedra have, in most cases, nearly identical coordination numbers in the ordered phase and the fluid phase close to the ordering transition. Because of this strong correlation, combining and allows for prediction of the assembly category expected for most cases. This Figure is reproduced from Ref. [31]. Copyright permission from American Association for the Advancement of Science (2012).
Figure 1.2 Tunable helical supracolloidal structures from a facile particle model. (a) Cartoon of patchy particle model used in the simulations. The top patch is a self-complementary patch while the other two patches are a pair of complementary patches. The relative directions of these patches are determined by angles and . (b) Right-handed double-stranded helix formed from patchy particles with patch direction of and . (c) Right-handed double-stranded helix with larger pitch and radius than those in (b), where the parameters are set as . (d) Left-handed double-stranded helix formed from patchy particles with patch direction of and . In (b)–(d), the building blocks and the top and side views of the helically supracolloidal structures and their geometrical representation are shown. This Figure is reproduced from Ref. 41.Copyright Permission from Nature Publishing Group (2014).
Figure 1.3 The rotational “rattle room” of patchy colloids. (a) Bonded contacts between triblock Janus spheres are maintained as long as the edges of the overlaid triangles cross the attractive dark gray patches. For a fixed particle configuration, the angular excursions that a particle can make without breaking any of its four contacts are independent of the corresponding excursions of neighboring particles. In two dimensions such a rotational entropy can therefore be calculated one particle at a time, and depends only on the bond angles and . (b) The hexagonal (left) and kagome (right) lattices have the same number of bonded contacts and thus the same energy per particle. However, for moderate densities, particles in a kagome lattice have less room to rattle by translation, yet more entropy from rotational and vibrational motions, than particles in a hexagonal lattice at the same density. The latter effect dominates, which favors the formation of the kagome crystal over the hexagonal. This Figure is reproduced from Ref. [15]. Copyright permission from Nature Publishing Group (2013).
Figure 1.4
A
-block entropic free energy contribution − (where is the conformational entropy of the block) per polymer chain, for large particles (, gray curve) and small particles (, black curve), as a function of . This Figure is reproduced from Ref. [56]. Permission from American Association for the Advancement of Science (2001).
Figure 1.5 The dependence of the excess entropy per bead, , calculated from the pair correlation function between -block beads. The inset diagrams (a)–(c) show the position of Janus nanoparticles and the organization of the stiff blocks around them in response to the increase of the chain stiffness. This Figure is reproduced from Ref. [59]. Copyright permission from American Chemical Society (2015).
Figure 1.6 (a) Janus nanoparticles tethered with long and short ligand chains. (b) The random arrangement of nanoparticles at the fluid interface for the initial simulations. The gray plane surface represents the fluid interface whereas the fluid beads are not shown for clarity. The arrows indicate the lateral and vertical changes of the simulation box while applying mechanical pressure. (c)–(e) Top-view snapshots of the interfacial nanoparticle patterns where (c), (d) and (e). The dashed circles in (c) and (e) highlight the phase domains of the same nanoparticle component. This Figure is reproduced from Ref. [60]. Copyright permission from American Chemical Society (2014).
Figure 1.7 The role of information in top-down manufacturing and bottom-up self-assembly. Top-down manufacturing assembles simple building blocks into complex objects by using assemblers. This approach has been the standard strategy for manufacturing, but becomes increasingly difficult at the molecular and nanoscale. By contrast, the bottom-up self-assembly of simple (high-symmetry) components invariably leads to simple structures. The self-assembly of complex structures requires information-rich building blocks (informed components). This Figure is reproduced from Ref. [62]. Copyright permission from Nature Publishing Group (2015).
Figure 1.8 Cartoon of the coarse-grained model. (a) A model DNA chain for a sequence used in the experiments. The bead size for the dsDNA portion of the chain is , corresponding to a diameter of approximately 2 nm; the bead size for the ssDNA portion is , corresponding to an approximate diameter of 1 nm. (b) A model spherical GNP core. (c) An example of two SNA-GNPs: the particle on the left has an 8#nm gold core and 40 DNA chains; the particle on the right has a 10 nm core and 60 attached DNA chains. For more details, see Ref. [79], from which this Figure is reproduced. Copyright permission from American Chemical Society (2012).
Figure 1.9 Snapshots of the crystal structures obtained from MD simulations. Left to right: The ideal crystal cell, a simulation snapshot after the system fully crystallized and the averaged positions of the SNA-GNPs. For more details, see Ref. [79], from which this Figure is reproduced. Copyright permission from American Chemical Society (2012).
Figure 1.10 Kinetics in the self-assembling process of the helically supracolloidal structure. (a) The number-averaged degree of colloidal clusters as a function of self-assembling time . (b) Variation in the number of patch groups in the course of self-assembly for species shown in inserts. (c) Number-averaged distributions of colloidal clusters at different times. Error bars in (a) and (b) indicate standard deviation. This Figure is reproduced from Ref. 41. Copyright Permission from Nature Publishing Group (2014).
Chapter 2: Developing Hybrid Modeling Methods to Simulate Self-Assembly in Polymer Nanocomposites
Figure 2.1 (a) Equilibrium state of a nanorod-filled polymer gel in a good solvent (not displayed). Inset: enlarged view of initiator modified nanorod with one end functionalized with polymer chains and a cross-section of the hexagonal nanorod. (b) Snapshot of nanoclay-embedded polymer network at monomer conversion 95%. Inset top: schematic representation of the network. Inset bottom: detailed structure of the nanoclay particle.
Figure 2.2 Schematic of the elemental reactions in (a) standard FRP and (b) living radical copolymerization of monomer and cross-linker. An asterisk indicates an active radical. Open beads show unreacted or partially reacted species and filled beads show fully reacted species. , , , , , and are probabilities of the respective reactions (initiation, propagation with monomer, propagation with unreacted bifunctional cross-linker, cross-linking with partially reacted cross-linker with pendent functional group, termination through combination, and termination through disproportionation).
Figure 2.3 Flowchart of the reaction scheme in the computational model of radical polymerization. Details of the individual steps are described in the text.
Figure 2.4 (a) Monomer conversion (squares) and (circles) as a function of the number of simulation time steps. (b) Number average molecular weight (squares), weight average molecular weight (open circles), and polydispersity index (crosses) as functions of the monomer conversion. Error bars arise from averaging over four independent runs.
Figure 2.5 (a) Numbers of total polymer chains (squares), chains with active ends (circles), and non-reactive (crosses) polymer chains as functions of the monomer conversion. (b) Numbers of termination reactions, combination (downward triangles), and disproportionation (rhombuses) as functions of monomer conversion. Error bars arise from averaging over four independent runs.
Figure 2.6 Polymerization kinetics with varying (a, b) initiation probability and (c, d) total termination probability . Other reaction probabilities are the same as those in the reference case. (a, c) Dependence of on the simulation time steps. (b, d) Polydispersity index as a function of the monomer conversion. Error bars arise from averaging over four independent runs.
Figure 2.7 (a) Conversions and of monomer (solid lines) and of cross-linker (dashed lines) on reaction times during copolymerization with initial concentration ratios . (b) Evolution of the reduced degree of polymerization (RDP) with monomer conversion for a systematic variation of the initial concentration ratios of cross-linker to initiator , while fixing . (c) Comparison of DPD simulated gel points with ATRP experimental values.
Figure 2.8 (a) Initial configuration of the gel after the upper layer was cut off. (b–e) Snapshots of the system when the rods diffuse and extend into the outer solution, taken at the following times: (b) , (c) , (d) , and (e) . Numbers identify the rods that extend into the outer solution.
Figure 2.9 Time evolution of the center of mass of the four rods in the -direction. The rod numbers correspond to the respective rods in Figure 2.8.
Figure 2.10 Time evolution of the -coordinate of the center of mass of the rods for eight independent runs (total 32 rods) with: (a) random initial distribution (position and orientation) of rods and (b) rods that were initially aligned vertically and facing upward and have , which is the distance between the tip of the rods and the interface.
Figure 2.11 (a–e) Regrowth of the top layer at the following monomer conversions: (a) 0%, (b) 25%, (c) 50%, (d) 75%, (e) 96%. Initial monomer and cross-linker concentrations are and . Initiator density is 0.25, which corresponds to a concentration . (f–j) Top-down views corresponding to frames a–e.
Figure 2.12 (a) Dependence of conversions and of monomer (solid lines) and of cross-linker (dashed lines) on reaction times during copolymerization, where and are the respective concentrations of unreacted monomer and cross-linker at a given time. (b) Number density profiles of the newly formed gel at different monomer conversions. The black dashed line represents the corresponding number density profile of the original gel at .
Figure 2.13 Density profiles of the system (including original gel, new gel, rods, and grafted chains) at monomer conversion 95% with: (a) different initial monomer concentrations for initiator density on the rod surface () and for initial cross-linker concentrations varying with by fixing ; (b) with different initial densities ( varies from 0.09% to 0.54%) for and . Insets are the corresponding snapshots of the gel system in (a) with and , and in (b) with and .
Figure 2.14 Top-down view of the spatial distribution of the cross-links at monomer conversion 96% for the regenerated gel system with and (corresponding to the simulation in Figure 2.11e and 2.11j).
Figure 2.15 Fraction of the number of inter-rod cross-links () with respect to the total number of cross-links formed () as a function of initiator density and initial monomer concentration. Data are taken at cross-linker conversion of at least 96%. Error bars indicate the variations among four independent runs.
Figure 2.16 Position of center of mass of nanorods. Left: the position of the center of mass of the four nanorods is measured over time for different initial monomer concentrations in the solvent. Right: system snapshots for (top) and (bottom) at the end of the simulation. The simulation box changes from to after the cut to incorporate more materials for regeneration.
Figure 2.17 Formation of an inter-gel cross-link. The radical from an active end (indicated by an asterisk) is first transferred to the additional cross-linker. At this point, the additional cross-linker with the radical may now form another bond. An open circle indicates that the particle has not yet fully reacted. is the reaction probability between a monomer with the radical and an additional cross-linker.
Figure 2.18 (a) Comparison of system with no additional cross-linker (left) and with 5% additional cross-linker (right). (b) Spatial distribution of reacted cross-linkers, which is defined as a cross-linker that has formed at least three bonds (including the original and additional cross-linkers), in the direction. The results are averaged across four independent runs.
Figure 2.19 (a) (squares) and the number of active polymers (circles) as functions of reaction time steps during polymerization. The number of clay particles in the solution is . (b) Number of clusters in the simulation system as a function of monomer conversion for the system with . Error bars arise from averaging over four independent runs.
Figure 2.20 Effect of on the evolution of (a) and (b) the number of clusters. (c) Number of X-chains formed due to the combination of growing polymer chains as a function of the number of clay particles . Error bars arise from averaging over four independent runs.
Chapter 3: Theory and Simulation Studies of Self-Assembly of Helical Particles
Figure 3.1 Model helix made up of a chain of partially overlapping hard spheres. The orientation of the helix in space is univocally defined by the direction of its main axis and its two-fold symmetry axis .
Figure 3.2 Helical shapes studied, with radius ranging from 0.1 to 1.0, pitch ranging from 1 to 10, and constant contour length .
Figure 3.3 Image of a system of hard prolate ellipsoids in the nematic liquid crystal phase. The image was created with the program QMGA [43].
Figure 3.4 Comparison between numerical simulations (circles) and Onsager-like theory for perfectly parallel helices, in the plane screw-nematic order parameter–volume fraction. Lines are theory results using the II virial approximation (dashed), the II virial approximation with Parson–Lee correction (dash-dotted) and the III virial approximation (solid).
Figure 3.5 The nematic order parameter, , as a function of the volume fraction for helices with radius and pitch , whose morphology is displayed in the inset. The plot shows the results of NPT–MC simulations (filled circles) and those of Onsager theory with Parsons–Lee correction, see Equation (3.14) (line).
Figure 3.6 Polar order parameter, , as a function of the volume fraction for helices with radius and pitch (see inset). The symbols show the results of NPT–MC simulations; the lines are meant as a guide to eye. I(isotropic), N(nematic), (screw-nematic), (screw smectic A), (screw smectic B), C(compact). Phase labeling is reported in Table 3.1; C is a high density compact phase.
Figure 3.7 The dependence of translational order parameter on layer spacing for helices having , on either side of the –Sm transition. The maximum value of is taken as the smectic order parameter.
Figure 3.8 Cartoon showing the quantities used to define the hexatic order parameter , Equation (3.31). The gray circles are transversal sections of cylinders enclosing the helices and the thick short line inside each circle indicates the orientation of the helix vector. The bond angle is the angle between a reference axis (thick solid line) in a plane perpendicular to the director , and the projection of the interparticle vector into this plane (dashed line).
Figure 3.9 Correlation function as a function of the projection of the interparticle distance along the director , from MC simulations of hard helices with radius and pitch and , at different values of the volume fraction .
Figure 3.10 Pairs of helices in antiphase (left) and in phase (right).
Figure 3.11 Reduced pressure versus volume fraction for helices with and pitch (top) and (bottom). The corresponding shape of the helix is displayed in the insets. Different phases are identified by different grays and labeled accordingly.
Figure 3.12 Cartoon of a top view of two neighboring smectic layers (I) and (II) with different in-plane organizations. (left), (center) and (right).
Figure 3.13 Snapshots from NPT–MC simulations of helices with , (top) and , (bottom). Density increases on moving from left to right, displaying the sequence of phases. Different grays are given according to the projection of the local tangent to helices onto a plane perpendicular to the director [43].
Figure 3.14 Phase diagram in the volume fraction versus pitch plane for helices having . Symbols correspond to calculated points. Different phases are labeled as indicated in Table 3.1. The symbol is used for state points that in NPT–MC simulations were found in the nematic phase, and then their cholesteric pitch was determined by an Onsager-like theory.
Figure 3.15 Cholesteric pitch , calculated in state points indicated as in the phase diagrams in Figure 3.14 (, full circles), Figure 3.16 (, open triangles) and Figure 3.17 (, asterics). is the volume fraction and is the pitch of the helical particles (only values greater than 3 are shown).
Figure 3.16 Phase diagram in the volume fraction versus pitch plane for helices having . Symbols correspond to calculated points. Different phases are labeled as indicated in Table 3.1; C is a high density compact phase. The symbol is used for state points that in NPT–MC simulations were found in the nematic phase, and then their cholesteric pitch was determined by an Onsager-like theory.
Figure 3.17 Phase diagram in the volume fraction versus pitch plane for helices having . Symbols correspond to calculated points. Different phases are labeled as indicated in Table 3.1; C is a high density compact phase. The symbol is used for state points that in NPT–MC simulations were found in the nematic phase, and then their cholesteric pitch was determined by an Onsager-like theory.
Chapter 4: Self-Consistent Field Theory of Self-Assembling Multiblock Copolymers
Figure 4.1 Phase diagram of AB diblock copolymer melts, consisting of the stability regions of lamellar (L), cylindrical (C), BCC spherical (S), HCP spherical (), gyroid (G), and () morphologies.
Figure 4.2 Phase diagram of a melt of an AB comb copolymer which is composed of a B-block backbone and regularly spaced teeth of A-blocks. Here, is the number of segments of each tooth.
Figure 4.3 (Top) Morphologies self-assembled in the ABC linear triblock copolymers with and , discovered by the general spectral method of SCFT. (Bottom) Phase diagram of these morphologies.
Figure 4.4 Two-dimensional morphologies obtained by solving the SCFT equations for ABCA tetrablock copolymer melts in real space. Regions rich in A, B, or C monomers are shown in gray, black, and white, respectively. The interaction parameters are fixed as , and the compositions of the four panels are: (a) , (b) , (c) , and (d) , respectively.
Figure 4.5 Phase diagrams of AB diblock copolymer identified by the fourth-order pseudo-spectral method, extending to a strong segregation region of .
Figure 4.6 (Left) Candidate super-cylindrical morphologies produced by the pseudo-spectral method coupled with a special initialization scheme. (Right) Two-dimensional cross-section of phase diagram of ABC linear triblock copolymers with fixed and , showing the stability regions of super-cylindrical phases including sphere-on-cylinder (SC), double helices-on-cylinder (), triple helices-on-cylinder (), and perforated lamella-on-cylinder (PC). The superscript “a” of or denotes that the helical super-cylinders have an anti-chirality arrangement and possess half left-handed and half right-handed helices.
Figure 4.7 Design principle of multiblock terpolymers for binary soft mesocrystals. (a) Artificial macromolecular “atoms” (AMAs) self-assembled by linear ABC triblock copolymers pack into the CsCl crystal lattice. (b) AMAs formed by multiblock terpolymers can be programmed to assemble into a variety of crystallographic arrangements by tailoring the polymer architectures. Three possible paths are demonstrated to design series of mesocrystals with varying magnitudes and asymmetries of coordination numbers (CNs) by tuning the relative lengths among the B blocks while keeping the total B component fixed.
Figure 4.8 (a) Phase diagram of symmetric pentablock copolymers in the – plane for fixed . (b) Phase diagram of asymmetric pentablock copolymers in the – plane, where quantifies the asymmetric degree of the molecule.
Figure 4.9 Phase diagrams of miktoarm block copolymers with , 3, and 4 in (a), (b), and (c), respectively, from top to bottom. In Figure (a) of , the boundaries between the A15 phase and the BCC phase as well as the cylindrical phase without considering the -phase are plotted in dashed lines.
Chapter 5: Simulation Models of Soft Janus and Patchy Particles
Figure 5.1 Graphical representation of soft Janus particle models. (a) Soft one-patch Janus particle model. (b) Soft ABA-type triblock Janus particle model. (c) Soft BAB-type triblock Janus particle model.
Figure 5.2 Graphical representation of soft patchy particle models.
Figure 5.3 Comparison of the performance of the soft ABA-type triblock Janus particle model implemented in GALAMOST with the corresponding CPU code. The performance data are measured by time steps per second. All simulated systems have identical simulation settings with the number of ABA-type triblock Janus particles, , , (), , and .
Figure 5.4 Representative equilibrium superstructures self-assembled from soft one-patch Janus particles by properly tuning the Janus balance and the strength of attraction between attractive patches while keeping , , and .
Figure 5.5 Biomimetic supracolloidal helices self-assembled from soft one-patch Janus particles by tuning particle softness. (a) Single helices (, (), ). (b) Double helices (, (), ). (c) Bernal spirals described as a stack of face-sharing tetrahedra (, (), ).
Figure 5.6 Typical equilibrium structures self-assembled from soft ABA-type triblock Janus particles with different Janus balance while keeping , (), , and : thread-like structures (), network structures (), and pyrochlore lattice structures ().
Figure 5.7 Representative non-close-packed (ncp) and close-packed nanostructures self-assembled from soft BAB-type triblock Janus particles, including 2D single-layer hexagonal ncp (sh-ncp), square ncp (s-ncp), honeycomb-like ncp (hc-ncp), and hexagonal close-packed (hcp) arrays, and vesicles with sh-ncp and s-ncp surface lattices, at , , and .
Figure 5.8 (a) Graphene-like two-dimensional structure with a hexagonal honeycomb lattice self-assembled from soft three-patch particles with regular triangular arrangement of the patches. (b) Diamond lattice structure self-assembled from soft four-patch particles with regular tetrahedral arrangement of the patches. In the simulations, , (), , and .
Chapter 6: Molecular Models for Hepatitis B Virus Capsid Formation, Maturation, and Envelopment
Figure 6.1 A schematic representation of an HBV virion. The virus has a double-shelled structure containing an icosahedral nucleocapsid, about in outer diameter, coated with three interrelated lipoproteins (S, M, L) known as the HBV surface antigens (HBsAg). Inside the nucleocapsid is a partially double strained DNA in strong binding with the highly flexible and positively charged C-terminal domains of the capsid proteins.
Figure 6.2 HBV lipoproteins (S, M, L) share a common S domain with different N-terminal extensions. The S protein consists of 226 amino acid (aa) residues; it is mostly imbedded in the lipid bilayer and can be divided into four trans-membrane (TM) regions. The M protein is N-terminally extended from the S protein with the Pre-S2 domain (with 55 aa). The L protein carries an additional N-terminal Pre-S1 domain (gray), which consists of 108 or 119 aa depending on the genotype. The N-terminus of the L protein is anchored into the lipid bilayer by a 14-carbon chain of saturated fatty acid added by myristylation (shown as the open circle). Upon translational membrane insertion, the Pre-S1 and Pre-S2 regions of the L protein are initially located on the cytosolic side of the endoplasmic reticulum (ER) membrane with TM1 not being inserted in the membrane (L: i-preS). In HBV virions, about half of the L proteins translocate the Pre-S region onto the viral surface (L: e-preS). Reproduced from Ref. [17] with permission from Springer.
Figure 6.3 The structure of the N-terminal region of the HBV capsid protein (HBcAg 1–149) in its dimer form (i.e., capsomer). The capsid dimer has a T-shaped structure stabilized by strong hydrophobic attractions between -helical hairpins.
Figure 6.4 Schematic of the HBV replication life cycle. The viral entry into human hepatocytes is initiated by the Pre-S1 domain of HBsAg binding to a receptor that is normally involved in bile acid transport in the liver. After being transferred into the cytosol, the nucleocapsid is disassembled, leading to the release of the genomic material. Viral replication starts with the formation of a covalently closed circular DNA (cccDNA) in the nucleoplasm; the cccDNA templates RNA transcription and subsequently protein synthesis. Nucleocapsid (NC) assembly is driven by strong electrostatic interactions between capsid proteins and pregenome (pg) RNA. Inside NC, the polymerase (P) translates pgRNA first into a single-stranded (ss) DNA, and then a partially double-stranded (ds) (>50%) DNA. The matured capsid is then enveloped with surface proteins in the endoplasmic reticulum (ER) or migrates back to the nucleoplasm for cccDNA amplification.
Figure 6.5 A course-grained model for the arrangement of HBV capsomers on T3 and T4 capsids. (a) Each capsomer consists of two core proteins (HBcAg without CTD), depicted as three tangentially connected spheres with their radii determined from the protein crystal structure. (b) A lattice model for the arrangement of capsid proteins at a triangular facet of a T3 capsid. (c) The lattice model for a T4 capsid. In both cases, the major capsid holes are shown as hexagons and triangles.
Figure 6.6 A schematic representation of an HBV nucleocapsid containing CTD and RNA chains. Here, denotes the thickness of Region 2, and is the radius at the capsid interior wall. The electron-microscopy image of an HBV capsid is adapted from the literature [82].
Figure 6.7 The equilibrium constant for the formation of CTD-free empty T4 capsids. Symbols are experimental data [57] and lines are theoretical predictions. The salt concentration (M) is denoted beside each line.
Figure 6.8 The effect of van der Waals force on capsid stability. The symbols and the dashed line are the same as those shown in Figure 6.7, and the solid line represents without considering van der Waals interactions.
Figure 6.9 The free energy of encapsidation for CTD and RNA chains as a function of the thickness of the complex layer. The salt concentration (M) is denoted beside each line, and the minimum points are indicated by triangle symbols.
Figure 6.10 The encapsidation free energy for CTD and RNA chains in a T4 capsid () as a function of the salt concentration ().
Figure 6.11 Hydrophobic and electrostatic effects on the equilibrium constant of empty HBV capsids. The solid line corresponds to the theoretical results if there is a 10% reduction of the buried hydrophobic area at each dimer contact; the dotted line is for a 10% increase of the capsid charged area; and the dashed line is for an increase of one unit charge for each capsid protein. For comparison, also shown in this Figure are the equilibrium constants of wild type capsids from theory (WT, dash-dotted line) and experiment (circle points) at .
Figure 6.12 Schematic of RNA packaging in an HBV nucleocapsid. The optimal genome size can be determined from a partial encapsidation of the RNA chain that minimizes the thermodynamic potential of the entire nucleocapsid in an aqueous electrolyte solution. The unpackaged RNA fragments are subjected to degradation by intracellular nucleases.
Figure 6.13 Theoretical predictions (black open square) and experimental data (Exp. 1 [115], circle; Exp. 2 [141], triangle) for the optimal RNA length (kb or kilo-base) in the wild type (WT) and mutant HBV nucleocapsids.
Figure 6.14 Correlation between the RNA size and the net charge of C-terminal tails . The solid line was proposed by Belyi and Muthukumar [68].
Figure 6.15 Radial distributions of RNA (solid line) and CTD chains (dashed line) in WT and mutant HBV nucleocapsids [147]. Here,
r
stands for the radial distance from the capsid center; the perpendicular dashed lines indicate the position of the capsid shell. (f) The average number of amino acid (aa) residues from the CTD tails located outside the capsid wall for WT and mutants.
Figure 6.16 The accumulated number of CTD residues for empty (a) and RNA-containing (b) HBV capsids. The accumulated number is obtained from a radial integration of the density profiles for the linker-CTD residues and normalized by the total number of CTD tails . In other words, the -axis corresponds to the summed number of residues per CTD tail up to the radial distance from the capsid center. As increases, the summed value approaches 42, which is the total number of traced residues for each tail. The accumulated values for charged (+ and −) and neutral (·) residue groups are shown separately.
Figure 6.17 The ratio of the CTD residues exposed to the outer region of HBV capsids. The CTD residues exposed in the region are counted for both the empty capsid and nucleocapsid. Each residue is ranked from the tethering site (1) to the end segment (34).
Figure 6.18 Schematic representation of the CTD location. (a) A 2-fold capsid pore and a dimer of the capsid core protein [64]. (b) Six CTD tails in each 2-fold pore. (c) CTD tails of the empty capsid. Here, the tails are distributed both inside and outside the empty capsid through the 2-fold hole.
Figure 6.19 Effect of phosphorylation on the distributions of RNA and linker-CTD tails. (a) RNA distribution in phosphorylated (pCTD) and unphosphorylated (CTD) capsids. (b) The distributions of phosphorylated (pCTD) and unphosphorylated CTD tails.
Figure 6.20 The sequence of amino-acid (aa) residues in the Pre-S domain of the large HBV surface protein (L-HBs). (a) N-terminal residues (1–174 aa) of L-HBs. (b) 8-mer peptides (P1∼P12) used in MD simulations. (c) 4-mer peptides (P4) used for molecular docking.
Figure 6.21 Snapshots from MD simulation for P4 peptide binding to the HBcAG protein. (a) Initial conformation with P4 randomly placed around HBcAg. (b) The P4-core structure after of simulation.
Figure 6.22 Evolution of P4–HBc association. (a) Change of the distance between the averaged centers of the P4 peptide and two binding domains (130–134 aa; 135–139 aa) from HBcAg. (b) Root mean square distribution (RMSD) of the backbone atoms in the P4 peptide.
Figure 6.23 Docking of sub-fragment P4-3 onto HBcAg. Shown here are the docked conformation for P4-3 (101–103 aa of L-HBs) and the approximated binding energy.
Figure 6.24 Radial distributions of CTD chains at different stages of maturation. Density profiles of the CTD residues are compared for the mature (with dsDNA), immature (with ssRNA), and empty capsids. Perpendicular dashed lines show the location of the capsid wall.
Chapter 7: Simulation Studies of Metal–Ligand Self-Assembly
Figure 7.1 (a) 4,4-benzene-1,3-diyldipyridine (L) and (b) 4,4-bipyridine (bpy).
Figure 7.2 Pd(II) model using cationic dummy atoms (CaDA) and nanosphere. The Pd(II) CaDA model uses four identical dummy atoms, which are coplanarly attached to the Pd ions, and evenly transfers the atomic charge of the Pd(II) cation to the four dummy atoms.
Figure 7.3 Pyridine-capped tridentate ligand
1
.
Figure 7.4 Initial structure made by random placement of 24 Pd(II) and 48 ligand
1
models (which corresponds to four nanospheres) in the cubic simulation box with a volume of .
Figure 7.5 Snapshots after the 250 ns LD runs setting = 1.0 (left), 2.5 (center), 4.0 (right). An enlarged image of a completed nanosphere is additionally shown in the central case.
Figure 7.6 Time variations of the coordination numbers of the six Pd(II) within the completed nanosphere. Each of the six plots is vertically shifted for clarity. Snapshots at the simulation times of 50, 55 and 80 ns are additionally shown in the figure.
Figure 7.7 Time variations of the ligand exchange rates for the LD runs with = 1.0, 2.5 and 4.0. The broken line corresponds to the exchange rates of the six Pd(II) ions within the completed nanosphere (instead of all the Pd(II) ions) for the LD run with = 2.5.
Figure 7.8 Pyridine-capped banana-shaped bidentate ligands
a
,
b
and
c
.
Figure 7.9 a) Initial structure generated by random placement of 48 Pd(II) and 96 ligand
a
models in the cubic simulation box (volume = ). (b) Snapshot after a 500 ns LD run with and nanocages in the circle and square, respectively.
Figure 7.10 Enlarged view of the nanocage from the snapshot after the 500 ns LD run in Figure 7.9b.
Figure 7.11 Time variations of the coordination numbers of the six Pd(II) within the completed nanocage. Snapshots at the simulation time of 440 ns are additionally shown in the figure.
Figure 7.12 nanocage from the snapshot after the 500 ns LD run.
Figure 7.13 Cluster size distributions of the ligand
a
as a function of time. The vertical axis corresponds to the mass fractions.
Figure 7.14 (a) , (b) and (c) cages from the snapshot after the 1 s LD run shown in Figure 7.13.
Figure 7.15 Cluster size distributions of the ligand
b
as a function of time. The vertical axis corresponds to the mass fractions.
Figure 7.16 (a) Initial structure and snapshot after (b) 0.4 s, (c) 0.6 s and (d) 0.8 s of the [] system simulation viewing along the -axis (hydrogen atoms are omitted for clarity). The bounding box corresponds to the simulation box with 3-D periodic boundary conditions. The box sizes and shapes are different after 0.4 s.
Figure 7.17 Time evolution of the number of coordination bonds (black line) and number of zero- to four-coordinated (gray lines) Pd(II) and the simulation box volume (black dashed line) in the self-assembly process of the [] system. The dashed horizontal line at corresponds to the full-bonding limit of the total coordination bonds.
Figure 7.18 Snapshot after 0.8 s of the [] system simulation with (a) = 4.0 and (b) = 1.0 (hydrogen atoms are omitted for clarity).
Figure 7.19 (a) Initial structure and snapshot after (b) 1.0 s of the [] system simulation (hydrogen atoms are omitted for clarity). Scales (box sizes) are different in these snapshots.
Figure 7.20 (a) Time evolution of the number of coordination bonds (black line) and number of zero- to six-coordinated (gray lines) Ru(II) in the self-assembly process of the [] system. (b) Corresponding time evolution with the Ru–N distance enlargement.
Figure 7.21 Snapshot after (a) 0.4 s, (b) 0.6 s, (c) 0.8 s and (d) 1.0 s of the enlarged Ru–N distance [] system simulation (hydrogen atoms are omitted for clarity). Scales (box sizes) are different in these snapshots.
Figure 7.22 The 4,4-bpy center of molecular mass radial distribution functions averaged over the last 100 ps without (gray line) and with (black line) the Ru–N distance enlargement and with the Ru–N distance enlargement and also the smaller = 1.2 (light gray line).
Chapter 8: Simulations of Cell Uptake of Nanoparticles: Membrane-Mediated Interaction, Internalization Pathways, and Cooperative Effect
Figure 8.1 (a) The morphologies of the NP clusters in the interior of the lipid bilayer; from left to right, the NP diameter, , was set to , , and , respectively. (b) The potential of mean force (PMF) between the NPs as a function of NP distance at different membrane tensions and NP sizes. (c, d) The calculated PMF as the third NP approaches the fixed NP cluster formed by two identical NPs of at (c) a positive membrane tension of and (d) a negative membrane tension . Reproduced from Ref. [17] with permission from the American Physical Society.
Figure 8.2 A series of simulation snapshots of membrane-curvature-induced NP attraction. The times of the simulation snapshots are: (a) , (b) , (c) , (d) , (e) , and (f) , the last corresponding to roughly . Reproduced from Ref. [21] with permission from Nature Publication Group.
Figure 8.3 Endocytosis of two identical NPs. Two smaller NPs of were placed on a membrane of (a–c), while two larger NPs of were placed on a membrane of (d–f). (a, d) show the initial and final structure of endocytosis of two identical NPs. (b, e) show the evolution of the distance between two NPs. (c, f) show the evolution of NP positions along the membrane normal direction. The initial inter-NP distances are (a–c) and (d–f), respectively. Reproduced from Ref. [15] with permission from the American Chemical Society.
Figure 8.4 Four kinds of membrane responses to adsorption of NPs: (a) Receptor-mediated endocytosis, (b) adhesion of the NP on the membrane surface, (c) penetration of the NP into the membrane, and (d) NP-induced membrane rupture. Both top view and side view are displayed for each process. Reproduced from Ref. [18] with permission from the Royal Society of Chemistry.
Figure 8.5 (Upper) Endocytic pathway of a spherocylindrical NP. The NP takes a general laying-down-then-standing-up sequence during endocytosis. (Reproduced from Ref. [29] with permission from the American Chemical Society.) (Lower) Computer-simulated snapshots of the translocation of ellipsoids with vertical and horizontal starting orientations. (Reproduced from Ref. [14] with permission from Nature Publication Group.)
Figure 8.6 (Upper) Typical states of the interactions between a lipid bilayer membrane and a graphene nanosheet. (a
1
) Graphene-sandwiched structure for PG with side length . (b
1
) Hemisphere vesicle structure for PG with . (c
1
) Adhering to the surface of the membrane for double-oxidized graphene oxide (dGO) with , and (d
1
) lying across the membrane for dGO with . (Lower) A two-dimensional phase diagram characterizes the interrelated effects of graphene size and oxidization degree on the equilibrium states of the graphene nanosheets interacting with the lipid bilayer membrane. The various shading patterns mark the approximate regions of these states. Reproduced from Ref. [33] with permission from Elsevier.
Figure 8.7 (Upper) Schematic drawings of different surface modified NPs (hydrophilic ligand on the NP surface in white, and hydrophobic ligand in purple). (Lower) Typical configurations for the penetration of a 16-SNP during its penetration processes. Reproduced from Ref. [36] with permission from the Royal Society of Chemistry.
Figure 8.8 (Left) Calculated surface tension of lipid membranes, , as a function of area per lipid. The representative snapshots show three typical phases of G7 dendrimers interacting with a lipid bilayer membrane at different surface tensions: (a) penetration, (b) penetration and partial wrapping, and (c) full wrapping. (Right) A phase diagram from the simulations. As membrane tension decreases, different states, from penetration (no wrapping), partial wrapping, to full wrapping, appear. Reproduced from Ref. [42] with permission from the American Chemical Society.
Figure 8.9 Typical snapshots during different internalization processes. (a) Nanoparticles with a diameter of form close-packed aggregates before internalization. (b) Nanoparticles having a diameter of aggregate into pearl-chain-like arrangements. (c) Independent endocytosis occurs for nanoparticles having a diameter of . Both top view and side view are displayed. Reproduced from Ref. [15] with permission from the American Chemical Society.
Figure 8.10 Snapshot of tube-like structures formed from multiple NPs interacting with a lipid vesicle. (Reproduced from Ref. [48] with permission from the American Physical Society.) (Left) The NPs were initially outside, and (Right) inside the vesicle. (Reproduced from Ref. [50] with permission from the American Physical Society.)
Figure 8.11 Time sequence of typical snapshots of two rod-like NPs with a lower ligand density (Upper) and higher ligand density (Lower). Reproduced from Ref. [51] with permission from the Royal Society of Chemistry.
Chapter 9: Theories for Polymer Melts Consisting of Rod–Coil Polymers
Figure 9.1 A rod–coil block copolymer of total contour length . Basic segmental volumes are considered for lengths and on coil and rod blocks, respectively.
Chapter 10: Theoretical and Simulation Studies of Hierarchical Nanostructures Self-Assembled from Soft Matter Systems
Figure 10.1 (a) Representative hierarchical structures self-assembled from ABC linear triblock terpolymers. Adapted with permission from Ref. [49]. Copyright 2012 American Chemical Society. (b) Representative hierarchical structures self-assembled from ABC star terpolymers. Adapted with permission from Ref. [60]. Copyright 2013 American Chemical Society.
Figure 10.2 Stepwise mechanism of microphase separation of ABC star terpolymers. The left panel represents the density profiles of A, B and C components; the right panel schematically illustrates the disorder-to-order transition of ABC star terpolymers. Adapted with permission from Ref. [61]. Copyright 2014 Royal Society of Chemistry.
Figure 10.3 (a) Hierarchical patterns of linear-alternating multiblock copolymers. Adapted with permission from Ref. [67]. Copyright 2005 American Chemical Society. (b) Parallel and perpendicular lamellae-within-lamella structures of ternary multiblock terpolymers. Adapted with permission from Ref. [71]. Copyright 2010 American Chemical Society. (c) Lamella-, cylinder- and sphere-based hierarchical nanostructures of multiblock terpolymers. Adapted with permission from refs. [75]. Copyrights 2010 American Chemical Society. (d) Nanostructures with liquid crystalline phases from rod–coil multiblock copolymers. Adapted with permission from refs. [77]. Copyrights 2013 American Chemical Society.
Figure 10.4 (a) Structural hierarchy of comb-shaped supramolecular polymers in experiments. Adapted with permission from Ref. [37]. Copyright 1999 Wiley-VCH. (b) Hierarchical structures self-assembled from simplified coil–comb molecules. Reproduced with permission from Ref. [85]. Copyright 2009 American Chemical Society. (c) Hierarchical self-assembly of AB diblock copolymer/C homopolymer blends via reversibly interactions. Adapted with permission from Ref. [93]. Copyright 2011 American Chemical Society.
Figure 10.5 (a) Self-assembled structures of block copolymers in poor solvent. Adapted with permission from Ref. [100]. Copyright 2003 American Chemical Society. (b) Prenucleation strategy to achieve hierarchical vesicles formed by amphiphilic diblock copolymers. Adapted with permission from Ref. [105]. Copyright 2006 American Chemical Society. (c) Hierarchical vesicles self-assembled from graft copolymers in backbone-selective solvent. Adapted with permission from Ref. [107]. Copyright 2013 Royal Society of Chemistry. (d) Periodic ordered structures hierarchically self-assembled from diblock copolymer stars. Adapted with permission from Ref. [108]. Copyright 2007 American Physical Society.
Figure 10.6 (a) Multicompartment micelles from ABC linear triblock terpolymers in A-selective solvent. Reproduced with permission from Ref. [116]. Copyright 2011 American Chemical Society. (b) Multicore micelles self-assembled from ABC linear triblock terpolymers in B-selective solvent. Adapted with permission from Ref. [117]. Copyright 2011 Royal Society of Chemistry. (c) Self-assembled nanostructures of ABC star terpolymers in A-selective solvent. Adapted with permission from Ref. [121]. Copyright 2009 American Chemical Society.
Figure 10.7 Self-assembly mechanisms of ABC star terpolymers in selective solvent. The upper and lower images correspond to the cases of polymers with short and long lengths of solvophilic arms, respectively. Adapted with permission from Ref. [122]. Copyright 2012 Royal Society of Chemistry.
Figure 10.8 (a) Co-assembly behaviors of AB and BC block copolymer blends in A- and C-selective solvent. Adapted with permission from Ref. [123]. Copyright 2008 American Chemical Society. (b) Aggregate morphologies of ABC star terpolymer/AB diblock copolymer mixtures in dilute solutions. Reproduced with permission from Ref. [125]. Copyright 2007 American Chemical Society. (c) Novel multicompartment micelles self-assembled from blends of ABC linear triblock terpolymers and ABC star terpolymers. Adapted with permission from Ref. [127]. Copyright 2007 American Chemical Society. (D) Novel hierarchical nanostructures from cooperative self-assembly of rigid homopolymer/rod–coil diblock copolymer mixtures. Adapted with permission from Ref. [130]. Copyright 2015 Nature Publishing Group.
Figure 10.9 (a) Hierarchically ordered structures formed by block copolymer/bidisperse nanoparticle mixtures. Adapted with permission from Ref. [137]. Copyright 2002 American Physical Society. (b) Hierarchical nanocomposites formed by directed co-assembly of block copolymer/nanoparticle blends in chemical templates. Adapted with permission from Ref. [141]. Copyright 2008 American Physical Society. (c) Lamellae-within-lamella structures obtained from linear-alternating multiblock copolymer/nanoparticle mixtures. Reproduced with permission from Ref. [142]. Copyright 2009 American Chemical Society.
Figure 10.10 (a) Self-assembled structures of nanoparticles tethered by flexible polymer tails in dilute solution. Adapted with permission from Ref. [147]. Copyright 2015 American Chemical Society. (b) Hierarchical self-assembly of diblock copolymer functionalized nanoparticles in selective solvent. Reproduced with permission from Ref. [148]. Copyright 2014 American Institute of Physics.
Chapter 11: Nucleation in Colloidal Systems: Theory and Simulation
Figure 11.1 The equation of state (solid line) and the corresponding conveniently shifted and scaled Helmholtz free energy density (thick dashed line) as a function of density for a system of van der Waals fluid at temperature . The gas–liquid phase coexistence densities are and , respectively.
Figure 11.2 Nucleation of carbon dioxide bubbles around a finger.
Figure 11.3 Illustration of the nucleation of phase in a metastable phase .
Figure 11.4 Gibbs free energy of a cluster as a function of the cluster radius according to the classic nucleation theory where and are the critical cluster and the height of the free energy barrier, respectively.
Figure 11.5 Top: Comparison between the – plane (left) and the – plane (right) for the Lennard-Jones system in three different crystalline structures and in the liquid phase. Each point corresponds to a particular particle, where 2000 points from each structure were chosen randomly. Bottom: The – plane (left) and the – plane (right).
Figure 11.6 The cluster size () as a function of time in MC cycles for a random selection of clusters that start at the top of the nucleation barrier.
Figure 11.7 The mean squared deviation (MSD) of the cluster size as a function of time in MC cycles. The cluster size has been measured every cycle and averaged over 100 cycles to reduce the short-time fluctuations. The slope of this graph is twice the attachment rate (Equation 11.41).
Figure 11.8 Schematic illustration of the FFS method. An ensemble of branched transition paths is generated simultaneously by firing trial runs from randomly chosen configurations at each interface in turn.
Figure 11.9 The cluster size as a function of time in MC cycles for four random trajectories at pressure starting with a cluster size of using kinetic MC simulations with step size and measuring the order parameter every MC steps.
Figure 11.10 (a) Crystal nucleation rates obtained from computer simulations for systems of monodisperse and polydisperse hard spheres as a function of packing fraction , and its comparison with experimental measurements. This Figure is reproduced from Ref. [3], and the Ref.2, 3 and 4 in the Figure correspond to Refs. [36, 37] and [38] in this chapter, respectively. (b) Snapshot of a cross-section of a critical nucleus of a hard-sphere crystal at a liquid volume fraction .
Figure 11.11 A comparison of the crystal nucleation rates of hard spheres as determined by the three methods described in this chapter, FFS, US and MD, with the experimental results from Refs. [36–38] and theoretical results from Ref. [3, 5].
Figure 11.12 Fraction of particles identified as either FCC or HCP, respectively, in clusters produced via molecular dynamics (MD), forward flux sampling (FFS) and umbrella sampling (US) simulations as a function of cluster size . All three methods agree and find the pre-critical clusters are predominately FCC.
Figure 11.13 Typical configurations of critical nuclei for the plastic crystal nucleation of hard dumbbells with aspect ratios (a), 0.15 (b), and 0.3 (c) at supersaturation . The dark gray particles are FCC-like, the light black particles are HCP-like particles, while the light gray particles are undetermined. Here is the center-to-center distance between two spheres of diameter in the dumbbell.
Figure 11.14 A cross-section through the center of several nuclei at the induction time. The light gray particles are face-centered cubic ordered while the dark gray particles are hexagonally close-packed ordered. (a) A nucleus possessing a single dominant stacking direction; (b) a multiply twinned nucleus orientated with its main five-fold axis into the page.
Figure 11.15 Typical configurations of the crystal structures for linear hard-sphere chains with chain length (a) and for ring-like polymers with (b) and (c). Only crystalline spheres are shown here. The black and gray spheres are HCP-like and FCC-like particles, respectively.
Figure 11.16 (a) Typical phase diagram of a molecular substance with a relatively long-range attractive interaction. This phase diagram corresponds to the Lennard-Jones 6-12 potential (solid curve in insert). This Figure is reproduced from Ref. [48] with permission of the American Physical Society. (b) Typical phase diagram of colloids with short-range attraction, which is shown as the solid line in the inset [49].
Figure 11.17 Contour plots of the free energy landscape along the path from the metastable fluid to the critical crystal nucleus for a system of spherical particles with short-range attraction. (a) The free energy landscape well below the critical temperature (). The lowest free energy path to the critical nucleus is indicated by a dashed curve. This curve corresponds to the formation and growth of a highly crystalline cluster. (b) As (a), but for . Here, and are the numbers of particles in the dense cluster and crystalline nucleus, respectively.
Figure 11.18 Phase diagram of hard spherocylinder in the (density) versus (aspect ratio), representation from Ref. [57], where I, P, N, Sm, S denote isotropic, plastic crystal, nematic, smectic and crystal phases respectively. Inset: illustration of a hard spherocylinder.
Figure 11.19 (a) Configurations for spontaneous crystal nucleation from a typical molecular dynamics trajectory at and , 1000 and 3000 (from left to right). Isotropic-like particles are drawn 10 times smaller than their actual size. (b) Gibbs free energy as a function of the number of rods in the crystalline cluster at pressure , 7.2 and 7.4. Inset: a typical configuration of a critical cluster () at .
Figure 11.20 Gibbs free energy of a hard spherocylinder fluid with as a function of the number of particles in the biggest cluster , as calculated by umbrella sampling MC simulations at pressures , 2.90, 2.95 and 3.00. Inset: a typical snapshot of the system with large clusters.
Figure 11.21 Positional (top) and orientational (bottom) structure factor of hard rods with at varying . The inset shows the pressure dependence of the orientation correlation length . The dashed line is the power-law fitting with exponent .
Figure 11.22 (a) A CsCl crystal formed by positive (radius 1.08 m) and negative (0.99 m) poly(methyl methacrylate) (PMMA) spheres. (b) -type crystal formed by positive (radius 0.36 m) and negative (1.16 m) PMMA particles. (c) NaCl-type crystal formed by charged (radius 1.16 m) and uncharged (0.36 m) PMMA particles. (d) Theoretical phase diagram of oppositely charged colloids with size ratio based on the screened Coulomb energy, where and are the radius of the oppositely charged colloids. is the charge ratio , and is inverse Debye screening length of the system.
Figure 11.23 Top: Composition of the largest crystalline cluster as a function of the order parameter for a binary mixture of gray (species 1) and black (species 2) hard spheres with equal diameter as obtained from umbrella sampling simulations at pressure . The steady-state distribution was obtained by using FFS with . Bottom: Contour plot of the two-dimensional free energy landscape as a function of and .
Chapter 12: Atomistic and Coarse-Grained Simulation of Liquid Crystals
Figure 12.1 (a) Structure of the 8CB molecule along with its phase behaviour as a function of temperature. Other molecules belonging to the same family as 8CB: (b) 5CB, (c) 6CB and (d) 7CB.
Figure 12.2 (a) Initial system for the mixture of 8CB and trans-7AB. (b) Final system for the same mixture. The 7AB molecules are shown separately for clarity. Mass density profiles along the layer normal for 8CB (open circles) and 7AB (filled circles) averaged over the last of (c) trans-7AB and (d) cis-7AB simulations. Mass densities of 7AB have been multiplied by a factor of 10 for clarity. The solid curves are fits to a low-order Fourier expansion. One notes that trans-7AB is preferentially incorporated into the 8CB layers; cis-7AB exhibits a deep minimum near the centre of the 8CB layer, indicating that it is excluded quite effectively from the 8CB layers.
Figure 12.3 Comparison between simulated (symbols) and experimental (continuous lines) mass densities for 8CB homologues. Reproduced from [41] with permission of John Wiley and Sons Ltd.
Figure 12.4 Molecular structures (left) and MM optimized geometries (right) of alkenic (a) di-, (b) tri- and (c) tetrafluoroterphenyls. The Figure is adapted from Ref. [44]. Reproduced with permission of the Royal Society of Chemistry.
Figure 12.5 Molecular structures (left) and MM optimized geometries (right) of (a) 2-[4-(butyloxy) phenyl]-5-(octyloxy)pyrimidine (2PhP) and (b) 2-[4-(tetradecyloxy)phenyl]-5-(tetradecyloxy) pyrimidine (PhP14).
Figure 12.6 Phase diagram of the binary system 2PhP/PhP14 at a pressure of . is the mole fraction of PhP14 in the mixture. Different phases exist for different ranges of . The nematic (N), phase exists at . The smectic A (SmA), phase exists at a very broad phase region. The smectic C (SmC), phase exists at lower mole fractions of PhP14, , and at higher mole fractions of PhP14, . Both isotropic (Iso), and crystal (Cr), phases cover the whole phase region at high and low temperatures respectively. Reproduced from [45] with permission of AIP.
Figure 12.7 (a) Snapshot of an SmA phase for a mixture with at (left). Schematic sketch (right). (b) Snapshot of an SmA phase for a mixture with at (left). Schematic version (right). (c) Snapshot of an SmC phase for a mixture with at showing the whole mixture (left), only the 2PhP molecules (centre) and only the PhP14 molecules (right). The PhP14 molecules are represented in white and 2PhP molecules in gray in the snapshots. In the schematic PhP14 molecules are represented by tall lines while 2PhP molecules are represented by short lines. Reproduced from [45] with permission of AIP.
Figure 12.8 Models for thin rods: (a) connected spheres and (b) hard spherocylinder with aspect ratio .
Figure 12.9 Overlap test for two rigid spheres of radius and .
Figure 12.10 Various cases for computing overlap criteria for spherocylinders: (a) parallel to each other, (b) on top of each other, (c) touching each other in parallel fashion.
Figure 12.11 Phase diagram of the hard spherocylinder of aspect ratio ranging from 0 to 100. AAA and ABC correspond to different arrangements of hexagonal crystal lattices. Reproduced from [53] with permission of AIP.
Figure 12.12 Phase diagram of bent-core liquid crystal as a function of opening angle obtained through MC simulation. The various phases shown include: isotropic fluid (I), nematic (N), polar smectic A (SmAP), smectic A (SmA), columnar (Col), polar crystal (XP) and crystal (X).
Figure 12.13 Final configurations from Monte Carlo simulations of bent-core molecules with opening angle as a function of reduced pressure (, where is the volume of a zigzag molecule and is the actual pressure). From left to right: isotropic phase , nematic and polar smectic A .
Figure 12.14 Configuration showing the polar smectic (SmAP) phase for an opening angle and . This phase disappears and becomes a non-polar smectic (SmA) at an opening angle of for the same value of . We have also shown the in-plane arrangement of the bow vector for both cases.
Figure 12.15 Columnar phase formed at an opening angle of .
Figure 12.16 (a) SmC tilt angle and (b) layer spacing as a function of for (left) and (right).
Figure 12.17 Equilibrated configurations for (a) and (b) for various values of . The rods are represented in grey and the bent-core molecules in black. Reproduced from [57] with permission of the American Physical Society.
Figure 12.18 Density profiles for rods (solid lines) and bent-core molecules (dashed lines) for for various values of . The bent-core density has been scaled by a factor of 30 for clarity. Reproduced from [57] with permission of the American Physical Society.
Figure 12.19 Phase diagram of zigzag-shaped molecules with aspect ratio as a function of opening angle and reduced pressure . The phase diagram shows the existence of various phases: isotropic liquid (I), nematic (N), smectic A (SmA), smectic C (SmC), columnar (Col), tilted crystal (XT) and crystal (X). The zigzag molecule is shown in the inset and the zigzag angle defined. Reproduced from [60] with permission of the American Physical Society.
Figure 12.20 Molecular structure of BHn-alkanoates.
Figure 12.21 Molecular structure of BHA7.
Figure 12.22 Molecular structure of discotic HAT6.
Figure 12.23 Molecular structure of HAT5 molecule.
Figure 12.24 Molecular structure of HBC derivatives.
Figure 12.25 (a) Molecular structure of the HBC/oligothiophene hybrid. (b) Molecular arrangement in the hexagonal columnar phase after of MD simulation.
Figure 12.26 A schematic representation of model chromonic and water molecules.
Figure 12.27 Simulation snapshot of the system after a simulation run starting from isotropic conditions for low (a) and high (b) concentration.
Figure 12.28 Simulation snapshot for the system with (a) model 1 and (b) model 2, after the simulation run starting from a pre-assembled configuration.
Figure 12.29 Molecular structure of SSY in NH hydrazone form (a) and hydroxy azo form (b).
List of Tables
Chapter 2: Developing Hybrid Modeling Methods to Simulate Self-Assembly in Polymer Nanocomposites
Table 2.1 Comparison among the gel points in ATRP experiments and DPD simulations
Table 2.2 List of DPD interaction parameters between different components in units of reduced temperature
T
*
Chapter 3: Theory and Simulation Studies of Self-Assembly of Helical Particles
Table 3.1 Summary of the different phases identified in the MC simulations of hard helices, along with the corresponding order parameters
Chapter 6: Molecular Models for Hepatitis B Virus Capsid Formation, Maturation, and Envelopment
Table 6.1 Free energy of self-assembly for T3 and T4 capsids
Table 6.2 The binding energy (kcal/mol) for the core protein–peptide interactions. Here, Δ
U
VDW
is the van der Waals energy; Δ
U
Coul
is the electrostatic energy; Δ
W
PB
and Δ
W
np
are the polar and non-polar contributions to the solvation energy; Δ
E
tot
is the total interaction energy, which is the sum of Δ
E
polar
and Δ
E
np
;
T
Δ
S
is the entropic energy term; Δ
G
is the binding free energy
Chapter 7: Simulation Studies of Metal–Ligand Self-Assembly
Table 7.1 Calculated metal–ligand binding energy (kcal/mol)
Chapter 9: Theories for Polymer Melts Consisting of Rod–Coil Polymers
Table 9.1 Theoretical studies of rod–coil (RC) diblock copolymers
Chapter 12: Atomistic and Coarse-Grained Simulation of Liquid Crystals
Table 12.1 Comparison of experimental densities and heat of vaporization with simulated numbers using LCFF for several standard liquid fragments [31]
Table 12.2 Comparison of N–I transition temperatures from simulation and experiments.
n
= 0, 1 and 2 correspond to the first three homologues of the phenyl alkyl-4-(4
′
-cyanobenzylidene)aminocinnamate series. Table taken from [38]. Reproduced with permission of John Wiley and Sons Ltd
Table 12.3 Phase sequence and transition temperature for the nCB families of liquid crystals [41]
Table 12.4 Force constants resulting from MD simulations. Reproduced from [62] with permission of the American Chemical Society
