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We investigate the self-organization of dipolar spheres into polymer chains as a fundamental model of the self-assembly of particles having anisotropic interparticle interactions. Our study involves a combination of modeling with vertically vibrated magnetic beads simulating a quasi-two-dimensional fluid at equilibrium and corresponding Monte Carlo...
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... i is the number of particles in a given cluster and N i is the number of chains of length i . In Fig. 3, we show L as a function of T for the experiments ͑ a ͒ and the simulations ͑ b ͒ for a range of C . We find a family of curves describing the general increase of L upon cooling. At higher C , the increase in L ͑ T ͒ occurs faster as T is lowered, reflecting the concentration dependence of the PT. The curves in Fig. 3 notably have a similar shape, and it is natural to seek a reduced variable description. Previous simulations of the PT in the Stockmayer fluid have demonstrated that a universal reduced variable description of the T dependence of L can be obtained by expressing T relative to its value at the PT, T , defined by an inflection point in . We also found that L at this temperature is nearly universal, taking a value 2.1 regardless of C . This constancy of L is predicted also by the analytic theory of equilibrium polymerization ͓ 11 ͔ . We checked this relation for the simulated analog of our experimental system and found that L at T ⌽ again lies in a narrow range for the concentrations we consider. This universality in the magnitude of L at T ⌽ sug- gest that we should similarly be able to reduce the scatter in the measurements in Fig. 3 by simply normalizing T by the temperature where L = 2, to obtain an approximate equation of state description of the T dependence of L . This procedure is motivated by the comparatively noisy nature of our ⌽ data. We see from the inset in Fig. 3 ͑ a ͒ that this procedure indeed reduces the scatter considerably and the reduction is quite good for the approach to the PT temperature T → T ⌽ where L = 2. Note that the simulation data in the inset of Fig. 3 ͑ b ͒ exhibits a tendency to saturate to a finite value, which is simply the number of particles in the system. This feature is a finite size effect and the growth of L at low T is apparently unbounded in the thermodynamic limit, as found before for the 3D Stockmayer fluid ͓ 1 ͔ . The chains in the experiment also visually exhibit a saturation of chain length to a size corresponding to the number of particles in the system at low T , but the particle tracking algorithm cannot follow these large clusters reliably, giving rise to the residual scatter in the inset of Fig. 3 ͑ a ͒ and the absence of a discernible plateau. Nevertheless, the inset of Fig. 3 ͑ a ͒ clearly indicates the sharp rise in L at lower T and the significance of L ͑ T ⌽ ͒ = 2 in defining a reduced variable description of these observations. We also note that our determination of ⌽ ͑ T ͒ below is consistent with L ͑ T ⌽ ͒ Ϸ 2 within experimental uncertainty in our experimental system and we could equally as well have reduced our L data by T ⌽ values determined directly from the inflection point of ⌽ . In order to directly compare T ⌽ in our experiments and simulations, we rescale our experimental T using the expression ̃ T = AT + B where A and B are constants to be determined. This method of rescaling is consistent with the thermal relation for vertically vibrated systems described in earlier work ͓ 7 ͔ . A and B are determined from a linear best fit to a plot of the T ⌽ values from the experiment ͑ at C = 0.020, 0.026, 0.039, 0.052, 0.078, and 0.104 ͒ vs the T ⌽ values from the simulation at similar C . For our experiments we find: A = 13.8 and B = 0.151. Figure 4 shows the PT temperatures, T ⌽ and ̃ T ⌽ ͑ ̃ T ⌽ are the rescaled experimental T ⌽ values ͒ , as a function C for the experiment ͑ filled circles ͒ and the simulation ͑ open circles ͒ . It has been shown that for living polymerization, the transition temperature is related to the concentration of monomers through the Dainton-Ivin ͑ DI ͒ equation ͓ 13 ͔ : T ⌽ = , 2 ⌬ s p + k ln C where ⌬ h p is the change in enthalpy upon polymerization, ⌬ s p is the change in entropy upon polymerization, and k is Boltzmann’s constant. Dudowicz et al. ͓ 1 ͔ have shown that the DI equation also approximately describes T for freely associating monomers, as in our measurements and simulations. In Fig. 4, we also show a fit of the DI equation to the simulation data, where ⌬ h p = −2.13± 0.19, and ⌬ s p / k = −8.5± 1.0. A similar fit to the experimental data gives ⌬ h p = −2.20± 0.34, and ⌬ s p / k = −8.9± 1.8. Note that T ⌽ and ̃ T ⌽ increase with increasing C , in a similar fashion for both the experiment and the simulation. Also, it has been shown in the PT observed in the Stockmayer fluid ͓ 13 ͔ that ⌬ h p is equal to the minimum of the interaction potential between two monomers, U , in ideal equilibrium polymerization. In the W.L. case of was our simulations, supported by U min NIH = −1.8, Grant which No. is within R21-EB- 15% of 00328501. the fitted K.V.W. ⌬ h p . Previous would like work to on thank Stockmayer the NIST-NRC fluids in Post- 3D ͓ doctoral 1 ͔ have Research also exhibited Program close for agreement financial support. between U min and ⌬ h p estimated from the the DI equation. We have determined that a mechanically “thermalized” system of shaken hard spheres with embedded magnets exhibits self-assembly into linear polymer chains and have re- produced essential features of this system by MC simulations that faithfully model the experimental system. Our determination of the PT line from the inflection point of the order parameter ⌽ as a function of temperature coincides within numerical uncertainty in both the experimental and simulated systems and the results obtained are in accord with the analytic theory of equilibrium polymerization developed by Dudowicz et al. ͓ 1 ͔ . This is the first experimental system of dipolar particle fluids that has allowed the investigation of the reversible polymerization that occurs in these fluids and we expect this type of model to offer fundamental insights into both the dynamics and thermodynamics of self-assembly in biological systems where anisotropic interparticle interactions and geometrical confinement effects are prevalent. W.L. was supported by NIH Grant No. R21-EB- 00328501. K.V.W. would like to thank the NIST-NRC Post- doctoral Research Program for financial ...
Context 2
... i is the number of particles in a given cluster and N i is the number of chains of length i . In Fig. 3, we show L as a function of T for the experiments ͑ a ͒ and the simulations ͑ b ͒ for a range of C . We find a family of curves describing the general increase of L upon cooling. At higher C , the increase in L ͑ T ͒ occurs faster as T is lowered, reflecting the concentration dependence of the PT. The curves in Fig. 3 notably have a similar shape, and it is natural to seek a reduced variable description. Previous simulations of the PT in the Stockmayer fluid have demonstrated that a universal reduced variable description of the T dependence of L can be obtained by expressing T relative to its value at the PT, T , defined by an inflection point in . We also found that L at this temperature is nearly universal, taking a value 2.1 regardless of C . This constancy of L is predicted also by the analytic theory of equilibrium polymerization ͓ 11 ͔ . We checked this relation for the simulated analog of our experimental system and found that L at T ⌽ again lies in a narrow range for the concentrations we consider. This universality in the magnitude of L at T ⌽ sug- gest that we should similarly be able to reduce the scatter in the measurements in Fig. 3 by simply normalizing T by the temperature where L = 2, to obtain an approximate equation of state description of the T dependence of L . This procedure is motivated by the comparatively noisy nature of our ⌽ data. We see from the inset in Fig. 3 ͑ a ͒ that this procedure indeed reduces the scatter considerably and the reduction is quite good for the approach to the PT temperature T → T ⌽ where L = 2. Note that the simulation data in the inset of Fig. 3 ͑ b ͒ exhibits a tendency to saturate to a finite value, which is simply the number of particles in the system. This feature is a finite size effect and the growth of L at low T is apparently unbounded in the thermodynamic limit, as found before for the 3D Stockmayer fluid ͓ 1 ͔ . The chains in the experiment also visually exhibit a saturation of chain length to a size corresponding to the number of particles in the system at low T , but the particle tracking algorithm cannot follow these large clusters reliably, giving rise to the residual scatter in the inset of Fig. 3 ͑ a ͒ and the absence of a discernible plateau. Nevertheless, the inset of Fig. 3 ͑ a ͒ clearly indicates the sharp rise in L at lower T and the significance of L ͑ T ⌽ ͒ = 2 in defining a reduced variable description of these observations. We also note that our determination of ⌽ ͑ T ͒ below is consistent with L ͑ T ⌽ ͒ Ϸ 2 within experimental uncertainty in our experimental system and we could equally as well have reduced our L data by T ⌽ values determined directly from the inflection point of ⌽ . In order to directly compare T ⌽ in our experiments and simulations, we rescale our experimental T using the expression ̃ T = AT + B where A and B are constants to be determined. This method of rescaling is consistent with the thermal relation for vertically vibrated systems described in earlier work ͓ 7 ͔ . A and B are determined from a linear best fit to a plot of the T ⌽ values from the experiment ͑ at C = 0.020, 0.026, 0.039, 0.052, 0.078, and 0.104 ͒ vs the T ⌽ values from the simulation at similar C . For our experiments we find: A = 13.8 and B = 0.151. Figure 4 shows the PT temperatures, T ⌽ and ̃ T ⌽ ͑ ̃ T ⌽ are the rescaled experimental T ⌽ values ͒ , as a function C for the experiment ͑ filled circles ͒ and the simulation ͑ open circles ͒ . It has been shown that for living polymerization, the transition temperature is related to the concentration of monomers through the Dainton-Ivin ͑ DI ͒ equation ͓ 13 ͔ : T ⌽ = , 2 ⌬ s p + k ln C where ⌬ h p is the change in enthalpy upon polymerization, ⌬ s p is the change in entropy upon polymerization, and k is Boltzmann’s constant. Dudowicz et al. ͓ 1 ͔ have shown that the DI equation also approximately describes T for freely associating monomers, as in our measurements and simulations. In Fig. 4, we also show a fit of the DI equation to the simulation data, where ⌬ h p = −2.13± 0.19, and ⌬ s p / k = −8.5± 1.0. A similar fit to the experimental data gives ⌬ h p = −2.20± 0.34, and ⌬ s p / k = −8.9± 1.8. Note that T ⌽ and ̃ T ⌽ increase with increasing C , in a similar fashion for both the experiment and the simulation. Also, it has been shown in the PT observed in the Stockmayer fluid ͓ 13 ͔ that ⌬ h p is equal to the minimum of the interaction potential between two monomers, U , in ideal equilibrium polymerization. In the W.L. case of was our simulations, supported by U min NIH = −1.8, Grant which No. is within R21-EB- 15% of 00328501. the fitted K.V.W. ⌬ h p . Previous would like work to on thank Stockmayer the NIST-NRC fluids in Post- 3D ͓ doctoral 1 ͔ have Research also exhibited Program close for agreement financial support. between U min and ⌬ h p estimated from the the DI equation. We have determined that a mechanically “thermalized” system of shaken hard spheres with embedded magnets exhibits self-assembly into linear polymer chains and have re- produced essential features of this system by MC simulations that faithfully model the experimental system. Our determination of the PT line from the inflection point of the order parameter ⌽ as a function of temperature coincides within numerical uncertainty in both the experimental and simulated systems and the results obtained are in accord with the analytic theory of equilibrium polymerization developed by Dudowicz et al. ͓ 1 ͔ . This is the first experimental system of dipolar particle fluids that has allowed the investigation of the reversible polymerization that occurs in these fluids and we expect this type of model to offer fundamental insights into both the dynamics and thermodynamics of self-assembly in biological systems where anisotropic interparticle interactions and geometrical confinement effects are prevalent. W.L. was supported by NIH Grant No. R21-EB- 00328501. K.V.W. would like to thank the NIST-NRC Post- doctoral Research Program for financial ...
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Citations
... The two-state model predicts that L/L A should equal 2 at T p at which the T dependences of both Φ and s c exhibit an inflection point, in accordance with the observation of Ngai and co-workers 250,251 that the apparent activation energy near T c is normally about twice the high-T activation energy. In the Stockmayer fluid of dipolar particles 252 and tabletop measurements of driven magnetic particles thermalized by vertical shaking, 253 the average string length L has been observed to be generally near 2. In the GET, the crossover temperature T c of glass formation is identified by the temperature at which the T Macromolecules pubs.acs.org/Macromolecules Perspective dependence of Ts c has an inflection point, so we may expect a close connection between T p and T c . ...
We provide a perspective on polymer glass formation, with an emphasis on models in which the fluid entropy and collective particle motion dominate the theoretical description and data analysis. The entropy theory of glass formation has its origins in experimental observations relating to correlations between the fluid entropy and liquid dynamics going back nearly a century ago, and it has entered a new phase in recent years. We first discuss the dynamics of liquids in the high-temperature Arrhenius regime, where transition state theory is formally applicable. We then summarize the evolution of the entropy theory from a qualitative framework for organizing and interpreting temperature-dependent viscosity data by Kauzmann to the formulation of a hypothetical "ideal thermodynamic glass transition" by Gibbs and DiMarzio, followed by seminal measurements linking entropy and relaxation by Bestul and Chang and the Adam-Gibbs (AG) model of glass formation rationalizing the observations of Bestul and Chang. These developments laid the groundwork for the generalized entropy theory (GET), which merges an improved lattice model of polymer thermodynamics accounting for molecular structural details and enabling the analytic calculation of the configurational entropy with the AG model, giving rise to a highly predictive model of the segmental structural relaxation time of polymeric glass-forming liquids. The development of the GET has occurred in parallel with the string model of glass formation in which concrete realizations of the cooperatively rearranging regions are identified and quantified for a wide range of polymeric and other glass-forming materials. The string model has shown that many of the assumptions of AG are well supported by simulations, while others are certainly not, giving rise to an entropy theory of glass formation that is largely in accord with the GET. As the GET and string models continue to be refined, these models progressively grow into a more unified framework, and this Perspective reviews the present status of development of this promising approach to the dynamics of polymeric glass-forming liquids.
... At the macroscale, pioneering studies and even some toys have used enthalpic interactions based on magnetic dipole forces to assemble 3-dimensional (3D) structures (22)(23)(24). Such forces are long-ranged and strong enough to direct assembly, but, to date, only the simplest dipole configurations have been explored, and the programmable potential of this approach to assembly is largely untapped. ...
... The magnetic moments of the panels allow for using external magnetic fields to facilitate assembly, disassemble undesired configurations, and, at lower field strengths, actuate the assemblies in complex environments (33)(34)(35). Such capabilities highlight the high degree of control that is available in the assembly and function of magnetic handshake materials, which is a great advance over macroscale assembly platforms that only use 1 or 2 magnetic dipole interactions and are agnostic to sequence (22)(23)(24). ...
Significance
Programmable self-assembly of smart, digital, and structurally complex materials from simple components remains a long-standing goal of material science. Here, we propose an assembly platform where information is encoded into building blocks using arrays of magnets that induce specific binding. Similar to current state-of-the-art platforms, magnetic encoding can achieve controlled polymerization of lone panels, complementary binding of panel strands, and 3D assembly of panel nets. This platform, however, has several advantages over current methods in that the approach is scale-invariant, has tunable mechanical strength, and has high information capacity for programmable self-assembly. We envision that this approach will lead to the formation of structures that transmit information, act as mechanical elements, or function as machines across length scales.
... Theoretical modeling of one-dimensional polymerization predicts that this type of aggregation results in an exponentially polydisperse distribution of polymer lengths, with an average length that changes smoothly upon varying density and temperature. 18,[25][26][27][28][29][30][31][32][33] In dilute conditions, parameter-free theoretical predictions 14,18,[28][29][30] properly describe the assembly process even in the limit of strong association, when the average length is much larger than one. ...
We numerically investigate cooperative polymerization in an off-lattice model based on a pairwise additive potential using particles with a single attractive patch that covers 30% of the colloid surface. Upon cooling, these particles self-assemble into small clusters which, below a density-dependent temperature, spontaneously reorganize into long straight tubes. We evaluate the partition functions of clusters of all sizes to provide an accurate description of the chemical reaction constants governing this process. Our calculations show that, for intermediate sizes, the partition functions retain contributions from two different structures, differing in both energy and entropy. We illustrate the microscopic mechanism behind the complex polymerization process in this system and provide a detailed evaluation of its thermodynamics.
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Magneto‐elastic materials facilitate features such as shape programmability, adaptive stiffness, and tunable strength, which are critical for advances in structural and robotic materials. Magneto‐elastic networks are commonly fabricated by employing hard magnets embedded in soft matrices to constitute a monolithic body. These architected network materials have excellent mechanical properties but damage incurred in extreme loading scenarios are permanent. To overcome this limitation, we present a novel design for elastic bars with permanent fixed dipole magnets at their ends and demonstrate their ability to self‐assemble into magneto‐elastic networks under random vibrations. The magneto‐elastic unit configuration, most notably the orientation of end dipoles, is shown to dictate the self‐assembled network topology, which can range from quasi‐ordered triangular lattices to stacks or strings of particles. Network mechanics are probed with uniaxial tensile tests and design criteria for forming stable lightweight 2D networks are established. It is shown that these magneto‐elastic networks rearrange and break gracefully at their magnetic nodes under large excitations and yet recover their original structure at moderate random excitations. This work paves the way for structural materials that can be self‐assembled and repaired on‐the‐fly with random vibrations, and broadens the applications of magneto‐elastic soft materials.
We provide a perspective on polymer glass formation, with an emphasis on models in which the fluid entropy and collective particle motion dominate the theoretical description and data analysis. We first discuss the dynamics of liquids in the high temperature Arrhenius regime, where transition state theory is formally applicable. We then summarize the evolution of the entropy theory from a qualitative framework for organizing and interpreting temperature-dependent viscosity data by Kauzmann to the formulation of a hypothetical `ideal thermodynamic glass transition' by Gibbs and DiMarzio, followed by seminal measurements linking entropy and relaxation by Bestul and Chang and the Adam-Gibbs (AG) model of glass formation rationalizing the observations of Bestul and Chang. These developments laid the groundwork for the generalized entropy theory (GET), which merges an improved lattice model of polymer thermodynamics accounting for molecular structural details and enabling the analytic calculation of the configurational entropy with the AG model, giving rise to a highly predictive model of the segmental structural relaxation time of polymeric glass-forming liquids. The development of the GET has occurred in parallel with the string model of glass formation in which concrete realizations of the cooperatively rearranging regions are identified and quantified for a wide range of polymeric and other glass-forming materials. The string model has shown that many of the assumptions of AG are well supported by simulations, while others are certainly not, giving rise to an entropy theory of glass formation that is largely in accord with the GET. As the GET and string models continue to be refined, these models progressively grow into a more unified framework, and this Perspective reviews the present status of development of this promising approach to the dynamics of polymeric glass-forming liquids.
The influence of the monomer chemical structure and architecture on polymer glass formation has been long appreciated to be dominated by two principal molecular factors, namely, cohesive energy and chain stiffness. Here, we utilize molecular dynamics simulation, along with the analytic generalized entropy theory (GET), to compare and contrast the role of cohesive energy in determining the thermodynamic and dynamic properties of polymer melts with and without bending constraints at zero pressure. As a general finding, our simulation results support the predictions of the GET that the temperature dependence of basic thermodynamic properties, such as the reduced thermal expansion coefficient and isothermal compressibility, becomes universal when the temperature is scaled by the cohesive energy parameter regardless of the bending constraints. Our simulation results confirm that dramatic changes in the temperature dependence of the structural relaxation time and extent of cooperative motion caused by cohesive interaction strength can likewise be eliminated largely using the same scaled temperature in polymer melts without bending constraints, but this simple picture does not emerge when bending constraints are introduced, a result in general accord with the predictions of the GET. Moreover, the fine features predicted by the GET, which are associated with the nonlinear growth of characteristic temperatures and the reduction of fragility with increasing cohesive interaction strength, are also confirmed by our simulations for polymer melts with bending constraints. Going beyond the validation of the predictions of the GET, we show that our simulation results for the structural relaxation time and extent of cooperative motion conform to the string model of glass formation in both polymer melts with and without bending constraints having variable cohesive interaction strength. We then utilize the string model to understand phenomenological models of structural relaxation of recent interest in glass-forming liquids. The string model also allows us to examine the dependence of the enthalpy and entropy of the high-temperature activation free energy on cohesive interaction strength. We emphasize the role of the activation entropy in improving the quantitative predictive capacity of the GET.
Dipolar active particles describe a class of self-propelled, biological or artificial particles equipped with an internal (typically magnetic) dipole moment. Because of the interplay between self-propulsion and dipole–dipole interactions, complex collective behavior is expected to emerge in systems of such particles. Here, we use Brownian dynamics simulations to explore this collective behavior. We focus on the structures that form in small systems in spatial confinement. We quantify the type of structures that emerge and how they depend on the self-propulsion speed and the dipolar (magnetic) strength of the particles. We observe that the dipolar active particles self-assemble into chains and rings. The dominant configuration is quantified with an order parameter for chain and ring formation and shown to depend on the self-propulsion speed and the dipolar magnetic strength of the particles. In addition, we show that the structural configurations are also affected by the confining walls. To that end, we compare different confining geometries and study the impact of a reorienting ‘wall torque’ upon collisions of a particle with a wall. Our results indicate that dipolar interactions can further enhance the already rich variety of collective behaviors of active particles.
