N. It turned out that while for the uncharged diblock copolymer the product χcrN was almost independent of N, for the diblock copolymers with one charged block we observed a significant increase in λcrN upon increasing N. This can be attributed to the fact that the counterion entropy prevents the formation of ordered structures, and its influence is more pronounced for longer chains since they undergo the transition to ordered structures at smaller values of λ, when the electrostatic energy becomes comparable to bTk. This was supported by studying the ODT in diblock-copolymers with charged blocks and counterions cross-linked to the charged monomer units. The ODT for such systems was observed at significantly lower values of λ, with the difference being more pronounced at longer chain lengths N. The fact that the microphase separation is observed even at zero Flory–Huggins parameter can be used for the creation of “high-χ” copolymers: The incorporation of charged groups  can significantly increase the segregation strength. The diffusion of counterions in the obtained ordered structures was studied and compared to the case of a system with the same number of charged groups but a homogeneous structure; the diffusion coefficient along the lamellar plane was found to be higher than in any direction in the homogeneous structure.In this work, we investigated the phase behavior of melts of block-copolymers with one charged block by means of dissipative particle dynamics with explicit electrostatic interactions. We assumed that all the Flory–Huggins χ parameters were equal to 0. We showed that the charge- correlation attraction solely can cause microphase separation with a long-range order; a phase diagram was constructed by varying the volume fraction of the uncharged block and the electrostatic interaction parameter λ (dimensionless Bjerrum length). The obtained phase diagram was compared to the phase diagram of “equivalent” neutral diblock-copolymers with the non-zero χ-parameter between the beads of different blocks. The neutral copolymers were constructed by grafting the counterions to the corresponding co-ions of the charged block with further switching off the electrostatic interactions. Surprisingly, the differences between these phase diagrams are rather subtle; the same phases in the same order are observed, and the positions of the order-disorder transition ODT points are similar if the λ-parameter is considered as an “effective” χ-parameter. Next, we studied the position of the ODT for lamellar structure depending on the chain length  Microphase separated polymer systems have attracted great attention due to their wide range of possible applications . Despiteb-poly(methylbutylene) diblock-copolymer the degree of sulfonation of the charged block plays an important role in determining the system morphology. Varying the sulfonation degree between 0 and 44.7%, the morphologies were found to change from disordered to gyroid to lamellae to hexagonally perforated lamellae (HPL). However, it was noted that no morphologies other than the ones present on the phase diagram of uncharged diblock-copolymer  were foeference : Inverteeference .One of the most interesting and dynamically developing areas of polyelectrolytes studies is the investigation of the properties of poly(ionic liquids). In short, polyionic liquids are a special type of polyelectrolyte which carry an ionic liquid IL species in each of the repeating units . Such arAs for the theoretical and simulation work, there is still no more or less systematic description of the microphase separation in polyelectrolytes, which is especially apparent if one considers the phase behavior of neutral copolymers that is studied very well even for different polymer chain architectures. Moreover, due to the presence of several types of interactions and length scales, it is a complicated task to create a consistent theoretical description of such systems. One of the most recent and complete investigation of the microphsase separation in melts of diblock-copolymers with one charged block was carried out in the papers ,23. In tIn this work, we addressed the question of microphase separation in melts of diblock-copolymers with one charged block. We focused on the effect of the presence of charged groups and mobile counterions and therefore investigated the case of completely compatible blocks (Flory–Huggins parameter χ = 0). The behavior of such systems was compared to that of corresponding uncharged diblock-copolymers as well as copolymers with counterions grafted to the corresponding co-ions of the charged block. Finally, we studied the influence of the ordering in the system on the counterion diffusion.In this work we used dissipative particle dynamics with explicit electrostatic interactions as the simulation method. First, we give a brief description of the standard model without electrostatic interactions. Dissipative particle dynamics (DPD) is a version of the coarse-grained molecular dynamics adapted to polymers and mapped onto the classical lattice Flory–Huggins theory ,25,26,27i- and j-th beads is equal to:ijr is the vector between i-th and j-th bead, αβa is the repulsion parameter if the particle i has the type α and the particle j has the type β and cR is the cutoff distance. cR is basically a free parameter depending on the volume of real atoms each bead represents [cR is usually taken as the length scale, i.e., cR = 1.The soft-core repulsion between presents ; Rc is ui and j) are connected by a bond, there is also a simple spring force acting on them:K is the bond stiffness and l0 is the equilibrium bond length.If two beads ; it can be found elsewhere .D is the damping distance. This approach allows one to prevent the overlapping of oppositely charged species while keeping the exact form of the Coulomb potential at distances larger than D; the parameter D is essentially the effective bead size and the electrostatic interactions at smaller distances are not important for the system behavior. We used D = 0.65 which was show in Reference [To take into account the electrostatic interactions, we use the method described in Reference . The maieference  to be a N = 24 were considered; each monomer unit of the charged block carried a charge of +e, while the counterions had a charge of -e . Since in the classical DPD all the beads have the same size, this is a model of a diblock-copolymer with one block being poly(ionic liquid) with a large free ion. The volume fraction of the uncharged block in the system φ was varied from 5/19 to 11/13; the volume fraction was calculated as n is the number of counterions in the system equal to the number of monomer units in the charged block n = N-unchN. If not mentioned, all the Flory–Huggins interaction parameters in the system were equal to 0 and the simulation box size was taken to be 403. The strength of the electrostatic interactions was characterized by a dimensionless parameter bl is the Bjerrum length in the system; it was varied from 1 to 20. If we consider parameterization similar to that proposed by Groot [cR ≈ 0.7 nm, then λ = 1 would correspond to a polar medium with ε≈80 at room temperature, while λ = 20 – to a medium with ε ≈ 4 at room temperature.The melts (no solvent was added) of diblock-copolymers with one charged block with the length of by Groot  with Rc The dependence on the position of the order–disorder transition on the chain length was additionally studied for the chains in which the counterions were grafted to the corresponding co-ions of the charged block; all other parameters were the same as for the system above.a = αβ – aααa  and χ for the used parameters  in the case of homopolymers has the following form: χαβ = 0.253 ± 0.004Δaαβ; this expression was used in what follows.Finally, the behavior of copolymers with one charged block was compared to that of corresponding uncharged diblock-copolymers, which were constructed by grafting the counterions to the corresponding co-ions of the charged block with further switching off the electrostatic interactions The incompatibility between the beads constituting different blocks was expressed in terms of the Flory–Huggins parameter χ. Using the approach described in Reference , we foun3. Additionally, some systems were studied in simulation boxes with the size of 323 and 483 in order to confirm that the observed structures are in equilibrium.The box size in our simulations was equal to 40First of all, we studied the phase diagram for the diblock-copolymer chains with one charged block. The resulting phase diagram is presented in The diagram is rather similar to the conventional phase diagrams of uncharged diblock copolymers ,8,10 in ABN), where χAB is the Flory–Huggins parameter between the beads of different blocks, which was varied from 0.5 to 2.7, and N is the length of the main (linear) chain ; it is presented in AB) and the subsequent error in the order–disorder transition (ODT) position as well as the presence of the counterion entropy in the charged diblock-copolymer melt. The only marked difference is the presence of the order–order transition (OOT) from PL to L at φ = 0.41 observed in the charged system and absent in the uncharged one. The observed lamellar structures in the charged diblock-copolymers at φ = 0.41 and high λ were prone to the formation of large defects, which might indicate the close proximity of the PL/L transition or even the coexistence of the PL and L phases. The possible reason for such a transition will be discussed below. The dimensionless parameters λ and χAB seem to play the same role in the microphase separation phenomenon, which is logical: They are both inversely proportional to bTk. Therefore, the driving force for the microphase separation in the studied diblock-copolymers with one charged block is completely different from that in “classical” diblock-copolymers, but the phase behavior seems to be almost independent of the exact nature of it.Let us now compare the obtained phase diagram to the phase diagram of the corresponding neutral diblock-copolymer where the behavior is governed by the χ-parameter. In some sense, the polyelectrolyte block and its counterions can be considered as a linear block with one additional bead attached to each monomer unit ; the box size was increased to 80approach ; the posL~χ1/6 (with L being the structure period) is reproduced. The same scaling L~λ1/6 is observed for the diblock-copolymer with one charged block at bTk at the length scales of one monomer unit), at smaller λ the counterion entropy plays an important role and prevents the formation of ordered structures.We can see that for the uncharged copolymer, even though one of its blocks has additional beads attached to one of the blocks, the theoretical dependence for “classical” diblock-copolymers in the strong segregation regime  of L~χ1/In order to test this hypothesis, we studied the position of the ODT point for different chain lengths for both types of block-copolymers at φ = 0.5; the obtained dependencies are shown in N for the uncharged diblock-copolymer is linear . While the structure of the ionic domain could be somewhat different from that observed in the system with free counterions, the general physical reasons behind the microphase separation are the same. The case with φ = 0.5 was investigated; we found that for the chain length of ααa and/or the electrostatic damping distance D. However, the former greatly increases the probability of the system being caught in a kinetically trapped state even at low segregation strengths, while significant changes in the latter will decrease the accuracy of taking into account the electrostatic interactions [ααa = 150 at λ = 16. The obtained defective lamellar structure had the same period (defined by the position of the first peak on the static structure factor) as that depicted in It is worth noting that since the microphase separation occurs due to the effective attraction (in our case it is the charge-correlation attraction), the domains formed by the attracting beads  in DPD will inevitably have larger density compared to the other domains . For the chosen parameters, at large values of λ > 14 at φ = 0.5 we obtained that the ionic domains have ~6–7% higher number density compared the average system value of ρ = 3. The deviations can be made smaller (but not completely removed) by increasing the repulsion parameter between alike species ractions . We addiAs was mentioned in the introduction, copolymers with polyelectrolyte blocks are often considered as new-generation materials for ion-exchanging membranes. For such applications, the ion conductivity is a crucial characteristic. Ion conductivity is related to the diffusion of free ions; therefore, we studied the influence of the presence of ordered structures on the ion diffusion. To that end, we compared the mean-squared displacement (MSD) of the counterions in lamellar structures at φ = 0.6 to that in melts of copolymers in which the charged groups were distributed evenly, but their number as well as the chain length was the same. The latter copolymers did not form any ordered structures even at the highest studied λ; the results are presented in First of all, we can see that in all studied cases the MSD shows a linear relationship to time, which means that the counterions undergo normal diffusion. No stable ion pairs are formed, and the counterions are “shared” between all the charged monomers  like electrons in metals. Similar behavior was observed in globules formed by a single polyelectrolyte chain ,37, in wDue to the homogeneous structure of the melts of copolymers with evenly spaced charged groups, the movement of the counterions is slowed down by the presence of neutral regions in the system; in some sense, there is no direct pathway formed by charged monomer units for a counterion between two points at any given moment of time. At large values of λ the charged groups seem to form small clusters surrounded by the neutral monomer units, which restricts the movement of the counterions. This can be easily understood if one considers a copolymer with only a small fraction of charged groups; if these groups are distributed evenly (or randomly) and the system are homogeneous, the charged groups are essentially isolated from one another and the counterions form stable ion pairs with the monomer units. The diffusion is possible through the segmental movement and “hopping” of counterions between the ion pairs, but is it much slower than the diffusion in a microphase-separated state if all the charged groups form one block; in that case, the counterions can freely move within the domains formed by the ionic species.In this work we investigated the phase behavior of melts of block-copolymers with one charged block by means of dissipative particle dynamics with explicit electrostatic interactions; such systems are a model of, for example, diblock-copolymers with one block being ionic liquid. We assumed that the blocks are fully compatible, i.e., all the Flory–Huggins χ parameters were equal to 0. This way, we showed that the charge correlation attraction solely can cause microphase separation with long-range order; a phase diagram was constructed by varying the volume fraction of the uncharged block and the electrostatic interaction parameter λ. Conventional phases like lamellae, hexagonally perforated lamellae, and hexagonally packed cylinders were observed. The obtained phase diagram was compared to the phase diagram of corresponding neutral diblock-copolymers with non-zero χ-parameter between the beads of different blocks; one block of such copolymers consisted of “dumb-bell” monomer units to mimic the increase of the volume of each monomer due to the presence of counterions. Surprisingly, the differences between these phase diagrams are rather subtle; the same phases in the same order are observed, and the positions of the ODT points are similar if the λ-parameter is considered as an “effective” χ parameter. Therefore, the dimensionless parameters λ and χ seem to play the same role in the microphase separation phenomenon; the phase behavior seems to be almost independent of the exact nature of the driving force behind it.N. It turned out that while for the uncharged diblock-copolymer the product N, which is in line with the theoretical predictions for “classical” diblock-copolymers with symmetric blocks, for the diblock-copolymers with one charged block we observed a significant increase in λcrN upon increasing N. We suppose this can be attributed to the fact that the counterion entropy prevents the formation of ordered structures, and its influence is more pronounced for longer chains since they undergo the transition to ordered structures at smaller values of λ, when the electrostatic energy becomes comparable to bTk. This was supported by studying the ODT in diblock-copolymers with charged blocks and counterions cross-linked to the charged monomer units (thus forming dipole monomers). Indeed, the ODT for such systems was observed at significantly lower values of λ with the difference being more pronounced at longer chain lengths N. Therefore, we can speculate that in the limit of Next, we investigated how the presence of free counterions and their entropy affects the behavior of the system. To that end, we studied the position of the ODT for a lamellar structure depending on the chain length Finally, we studied the diffusion of counterions in the obtained ordered structures and compared it to the case of a system with the same number of charged groups but a homogeneous structure; the diffusion coefficient along the lamellar plane was found to be higher than in any direction in homogeneous structure. This can be attributed to the fact that in homogeneous structures the movement of the counterions is restricted by the presence of uncharged monomers surrounding the charged ones, while the formation of purely ionic phases in the case of ordered structures allow the counterions to move freely within the domains.It is worth noting that in this work  can significantly increase the segregation strength.


Modern electronic structure calculations are predominantly implemented within the super cell representation in which unit cells are periodically arranged in space. Even in the case of non-crystalline materials, defect-embedded unit cells are commonly used to describe doped structures. However, this type of computation becomes prohibitively demanding when convergence rates are sufficiently slow and may require calculations with very large unit cells. Here we show that a hitherto unexplored feature displayed by several 2D materials may be used to achieve convergence in formation- and adsorption-energy calculations with relatively small unit-cell sizes. The generality of our method is illustrated with Density Functional Theory calculations for different 2D hosts doped with different impurities, all of which providing accuracy levels that would otherwise require enormously large unit cells. This approach provides an efficient route to calculating the physical properties of 2D systems in general but is particularly suitable for Dirac-point materials doped with impurities that break their sublattice symmetry. Most of these tools are based on ab initio atomistic calculations of the electronic structure of materials often described within small unit cells that are periodically repeated to span the entire space. The use of such periodic boundary conditions (PBC) is the norm not only in the treatment of pristine crystalline materials. In fact, supercell calculations consisting of identical defect-embedded cells periodically arranged in space are also the standard for studying non-crystalline structures and are believed to capture the essential characteristics of disordered materials containing dopants2.Modern tools to calculate physical and chemical properties of materials have reached unprecedented accuracy levels5 based on Density Functional Theory (DFT)7, cell sizes must be sufficiently large to avoid the interaction between the periodically spaced defects. Problems are known to arise when convergence is very slow and reached only with cell sizes that make the calculations too demanding. Several attempts have been made to introduce finite-size supercell corrections in order to mitigate the computational costs of such calculations but they are primarily aimed at 3D materials and not necessarily at their 2D counterparts14.While the PBC approach permits the use of well-established computational packages available for materials properties simulations15. However, rather than accepting this feature as a hindrance, here we explore this peculiarity seen in many 2D materials to introduce a finite-size supercell correction scheme that reduces the computational costs and/or increases the accuracy levels of calculations in these materials by orders of magnitude. The goal of our formulation is to determine energy variation quantities associated to a host material being perturbed by the presence of an impurity. This is done via an analytic approach that gives the energetics of the perturbed system in a transparent functional form. This work is a generalization of a correction scheme we developed originally for carbon nanotubes14. Although carbon nanotubes are quasi-one-dimensional structures, they share some key characteristics of 2D materials as they can be thought of as rolled up graphene sheets.One peculiar feature of several 2D materials that may aggravate the slow-convergence problem even further is when impurity-induced Friedel oscillations become perfectly commensurate with the underlying lattice. In this case, the interaction between impurities that would otherwise vanish is enhanced and may retard the convergence of supercell calculations quite considerablyThe goal of our formulation is to determine energy variation quantities associated to a host material being perturbed by the presence of an impurity. This is done via an analytic approach that gives the energetics of the perturbed system in a transparent functional form. For the sake of illustration, the method will be presented for the case of a graphene sheet (host) doped by a single substitutional atom cf. Fig.  but it cj〉 centered at sites identified by the integer j. The perturbation potential λ) in the on-site potential at the impurity site (j = 0). The matrix elements ti,j = t when i and j label nearest-neighbour sites but vanish otherwise. As we shall see, the actual value of t is arbitrary and immaterial since we are primarily interested in establishing a functional form for the energy variation due to the presence of impurities. The simplicity of this single-orbital picture is in no way a limiting factor since it can be easily generalized to a multi-orbital representation which may also account for further nearest-neighbour couplings if necessary. Likewise, the premise that the introduction of a substitutional impurity does not impact the on-site energies of its neighbours is a consequence of the screening and can be easily relaxed without any qualitative change to our results and conclusions. Attention is drawn to the fact that this Hamiltonian is written in real space and describes a single impurity embedded in an ideally infinite host. This means that We start off with a generic Hamiltonian 1 associated with the alteration of the host material by a single substitutional impurity can be quantified usingE is the energy, f(E) is the Fermi-Dirac distribution function, and 1 is associated with the energy-band contribution to the total energy variation and is present in quantities like the binding energy and formation energy, for instance. Note that the expression above contains a sum over all sites 160,0 being the GF of the pristine host projected onto the impurity site. Although 1 is not position dependent. Indeed, equation  is in general oscillatory and obeys the following functional form C(x) ∝ cos(Qx)/xα, where Q and α are constants that depend on the geometry of the host’s Fermi surface and on the system dimensionality, respectively17.Following the same line of thought, we compute the energy cost associated with the introduction of a second impurity onto the host. In the case of a pair of identical impurities, the impurity potential becomes C(x), one can conjecture that the sum in equation . Therefore, the energetics of graphene with N impurities becomesN/N is the energy-cost per impurity and γ is a constant. Even though the individual terms in the summation may be substantially smaller than Δ1, they add up to a finite quantity when the oscillation of C(x) is suppressed. In this case, the interaction between impurities can certainly impact on the overall energy balance of the system, leading to ΔN/N ≠ Δ1.As a result of the oscillatory behaviour of equation  will alwC(x) is what happens when impurities are orderly distributed. Consider for example a graphene sheet divided into identical diamond-shaped cells of length L, all of which containing one substitutional impurity as shown in Fig. L−dependence of ΔN/N can be easily extracted by simply factoring 1/Lα out of each term in the sum of equation (β is a constant.One striking consequence of the absence of oscillations in equation . The eneC(x) which turns equation . However, this is not so in the case of some 2D materials for which the oscillatory character of C(x) is suppressed and, consequently, ΔN/N alone is not sufficient to give direct information about Δ1. Far from invalidating the supercell methodology, we argue that this feature can be used to accelerate the convergence of such calculations, thereby enhancing its accuracy. By making use of the functional form shown in equation  and Ni (top-adsorbed) atoms were the chosen dopants. We note that, while the lowest energy configuration for Ni on graphene is the hollow hexagonal site, it does not break sublattice symmetry which, as we shall see, turns out to be an essential ingredient for the extrapolation method to work. Thus for the purposes of this work, we consider this metastable structure to demonstrate that our technique is also effective with adsorbed impurities that break sublattice symmetries.While other interactions may play a role in the total energy calculations, they are not as long-ranged as the interaction captured by equation  into theequation  is likelequation , a plot L is varied from 4 to 9 or 10 lattice parameters a. The electronic structure simulations were conducted with ab initio DFT as implemented in SIESTA5 with the Generalized Gradient Approximation18 (PBE-GGA) for the exchange-correlation functional. The valence electrons are described by a double-ζ polarized (DZP) basis set, and norm-conserving pseudopotentials19 were used. The cutoff for the real space grid was set at 400, 800, or 1000 Ry, and a k-grid of up to 16 × 16 × 1 was used depending on the size of the unit cell. In order to prevent spurious interaction between image cells, a vacuum region of 30 Å in the transverse direction was introduced. All structures were fully relaxed using a conjugate gradient algorithm with residual forces in each component of every atom smaller than 0.01 eV/Å. For the specific case of adsorbed Ni, graphene was relaxed in its pristine form first and then the Ni atom was placed on it. The Ni was allowed to relax transversely but not the graphene sheet in order to reduce noise in the results.Figure ab initio methods can reproduce the energetic behaviour of equation (Epris(L) corresponds to the total energy of a pristine 2D host with supercell length L, Edef(L) is the energy of its counterpart containing the impurity, i removed (ni < 0) to the system, and finally μi is the chemical potential for the corresponding species. The chemical potentials of the N and B impurities were taken considering as reference molecules N2 and diborane (B2H6), respectively. For an adsorbed impurity, we consider another energy quantity, namely the binding energy,EH+A corresponds to the total energy of the host with the adatom, and EH (EA) is the total energy of the isolated host (adatom). We note that the functional form of both measures (Ef and Eb) is the same. Ultimately the choice of chemical potential is somewhat arbitrary and will only introduce a rigid shift on Ef as it is not dependent on the size of the system. The same is true for the energy of the isolated adatom. Their choice is significant for a quantitative description of the formation/binding energy, but will not interfere in the behaviour of the curve as a function of L.To check if equation , we consEf changes with the size of the unit cell. Note that in particular for the case of the N-doped system, convergence is rather slow and not reached for L < 10a, even though there is no direct coupling between images of the impurity atoms for Ef. Making use of the predicted functional form shown in equation  describing the interaction between impurities which, in turn, is linked with the existence of Dirac points. In other words, systems whose electronic structure possesses Dirac points22 or for which the corresponding Fermi surface lies on the edges of the Brillouin zone are most likely to display these features. Furthermore, another essential ingredient is that the perturbing potential associated with the impurity must impact sublattices asymmetrically. This latter requirement is obviously met by the introduction of substitutional impurities onto the host.Our finite-size formulation is by no means exclusive of graphene. To avail of this extrapolation method, certain ingredients are necessary. Obviously, one needs the equation . This isL is increased. However, very distinctive straight lines are visible when Ef is plotted on the right panels as a function of 1/Lα, from which we can easily extract the intercept values through extrapolation. Similarly, the discrepancy between the extrapolated values and those obtained with L ≈ 10a is significant and ranges at approximately 25 meV.To illustrate this point we demonstrate that the extrapolation technique works equally well with systems containing both of these necessary ingredients, namely the Dirac-point and the sublattice-asymmetry requirements. Silicene is one ideal candidate since it has Dirac points. The top and middle panels of Fig. Eb is plotted as a function of the rescaled quantity 1/Lα.Substitutional impurities are not the only type giving rise to sublattice-asymmetric perturbing potentials. Adatoms are known to adsorb to 2D materials at different locations of their atomic structure. Top-adsorbed impurities are those attached more strongly to one of the host atoms and should meet the requirement of sublattice asymmetry in the perturbing potential. We put this requirement to the test by calculating the binding energy of Ni impurities that were top-adsorbed to graphene cf. Fig. . The botα = 223. However, the values of α used to transform the energetic curves into straight lines are not the same and differ depending on the host, on the impurity and on the bonding geometry, as shown in Figs α is a consequence of the simplifying assumptions made in the derivation of equation  is no longer applicable when the Fermi level shifts from the Dirac point. Nevertheless, if the charge transfer and consequently the Fermi-level shift are sufficiently small, the infinite series that appears in the r.h.s. of equation  into a linear function, we constrain the fitting scheme to search for α in close proximity to the dimensionality-defined value of α = 2. This was the procedure carried out in every single case shown in Figs α and how it depends on the charge transfer between host and impurity is a challenging task that deserves attention of the community working in this field. Once established, it will provide accurate values of formation and binding energies from computations involving only very small unit cell sizes. Meanwhile, even without full knowledge of how α scales with the charge transfer, calculations can be immensely simplified by the extrapolation scheme presented here.The exact value of equation , i.e., Δentclass1pt{minimaab-initio calculations. Moreover, we demonstrate that these quantities may display very slow convergence rates when they are calculated through the usual procedure of increasing the size of unit cells for the specific case of systems that present a commensurability effect between the impurity-induced Friedel oscillations and the lattice. When that happens, prohibitively large unit cells may be needed to reach desirable accuracies. Rather than invalidating the methodology that is so prevalent in electronic-structure calculations of 2D materials, we argue that we can make the most of that slow convergence rate to introduce an extrapolation scheme that provides results with accuracy levels that would otherwise require enormously large unit cells. Furthermore, our expressions were derived with doped graphene sheets in mind but our approach is by no means exclusive of that material and has also been shown to work for other 2D structures with different types of dopants, namely substitutional and adsorbed impurities. Finally, although our conclusions were based on the DFT results used throughout this paper, any other computational tool implemented within the supercell representation is bound to display exactly the same behaviour and can be optimized through the proposed extrapolation scheme.In summary, we have derived a useful expression for the change in total energy due to the formation of an array of periodically spaced impurities in 2D materials. This expression captures how quantities like formation energy and binding energy scale with the size of the unit cell, which is so commonly used in supercell implementations of


Approximate computing is an emerging design paradigm that offers trade-offs between output accuracy and computation efforts by exploiting some applications’ intrinsic error resiliency. Computation of error metrics is of paramount importance in approximate circuits to measure the degree of approximation. Most of the existing techniques for evaluating error metrics apply simulations which may not be effective for evaluation of large complex designs because of an immense increase in simulation runtime and a decrease in accuracy. To address these deficiencies, we present a novel methodology that employs SAT (Boolean satisfiability) solvers for fast and accurate determination of error metrics specifically for the calculation of an average-case error and the maximum error rate in functionally approximated circuits. The proposed approach identifies the set of all errors producing assignments to gauge the quality of approximate circuits for real-life applications. Additionally, the proposed approach provides a test generation method to facilitate design choices, and acts as an important guide to debug the approximate circuits to discover and locate the errors. The effectiveness of the approach is demonstrated by evaluating the error metrics of several benchmark-approximated adders of different sizes. Experimental results on benchmark circuits show that the proposed SAT-based methodology accurately determines the maximum error rate and an average-case error within acceptable CPU execution time in one go, and further provides a log of error-generating input assignments. Modified ALLSAT Solver Algorithm                    L. The variables ER and AVG are initialized to zero at the beginning of the algorithm. The ALLSAT solver begins to iteratively find errors in the CNF file. The SAT solver knows that there are no more remaining errors and it can terminate the search, when the problem is no longer satisfiable (Line 3). The SAT solver then works on the CNF file until a satisfying solution (counterexample) is found (Line 4). This solution is presented as a total variable assignment CEX to all inputs, outputs and intermediate variables. It is therefore obvious to state that the value of the single output of the miter will be logic 1 for any satisfying variable assignment. However, for our intended purposes, many intermediate variables are not needed, and in fact, we only care about the original inputs and outputs of both circuits. As seen in Line 5, the assignments the SAT solver found to these relevant variables are extracted from the counterexample. The assigned input is added to the list of error-producing input assignments (Line 6). The two outputs corresponding to the outputs of the original exact and approximate circuits are also extracted. We consider the absolute difference between the integer representation of each value output1 and output2 and accumulate it in the variable AVG (Line 7). In Line 8, the variable ER is incremented to count error-producing input assignments. The SAT solver then updates its internal information before reiterating the entire process until termination (Line 9). This information may include storing intermediate variable implications it learned to avoid redundant processing, as well as taking note of the input assignment it found in the past loop iteration. This is done in order to eventually eliminate all error-producing input assignments and thus making the problem not satisfiable.A more detailed view of the modified ALLSAT solver (MOD-ALLSAT) used in Algorithm 1 can be seen in Algorithm 2. The MOD-ALLSAT algorithm takes a CNF file and a variable map as inputs and calculates error-rate (ER), average-case error (AVG), and optionally produces the list of counterexamples A clarifying analogy to this would be deleting the path that has led to the 1 node in the BDD representation so it cannot be followed again. The SAT solver does this iteratively until eventually the equation cannot be satisfied anymore. This means all errors have been detected. The ALLSAT solver is also special in that it does not start from scratch on every iteration (it learns from previous iterations).ER now contains the numerator of Eq . Then the algorithm finishes execution and outputs the error-rate in variable ER, average-case error in variable AVG, and optionally the list of error-producing inputs L. It is worth nothing that both errors are calculated in one go of the SAT solver and with the same auxiliary circuit (the approximation miter).After terminating the search, the variable or of Eq  for maxior of Eq . The errIn order to prove the correctness of the designed methodology, a naïve example was developed for testing. A simple exact ripple carry 4-bit adder and an “approximated” 4-bit adder were written in Verilog. The “approximated” adder was a normal ripple carry adder that added one to every value it calculated to artificially introduce errors. For example, in the approximated adder: 1+1 = 3, 2+2 = 5, etc.F (the ripple carry 4-bit adder), and the approximated circuit  G. The two circuits are 4-bit adders; therefore, they have 5 outputs each. Yosys is used to apply a miter that XORs each of the 5 outputs to the corresponding output in the other circuit and ORs the result of all the XORs (Line 1). The resulting circuit is produced from Yosys to be converted into a CNF file (Line 2) so it can be fed to the ALLSAT solver. The ALLSAT solver runs on this representation of the miter combining the two circuits (Line 3) and begins to extract all the errors (differences) between them.Following the steps of Algorithm 1, the inputs to the algorithm are two circuits: an exact one L) is 256 . Lastly, the average-case error is calculated by adding each of the differences generated by all input assignments and dividing by the number of possible assignments. Since the approximate circuit simply outputs the result of the exact adder plus one, then the difference in all errors is one. Therefore, the total accumulated difference is 256 , calculated in Line 7 of the algorithm. The average error rate is then 256/256 (Line 9) such as in Now, since the approximate adder was made to increment one to the result of the exact adder, this means that all input statements produced errors. Hence, the error-rate is 100% (Line 4) and the number of errors  is 7.5. The main bulk of the CPU time is spent on the SAT solving phase of the methodology, as the application of the miter has negligible processing time. As expected, the CPU time jumps significantly between the 8-bit adders and the 16-bit adders, because the search space grows exponentially with more inputs, making it more compute-intensive to find errors.The error rate numbers for all the approximate adders exactly matched the ones mentioned in , validatFrom the results, several approximate adders have the same error-rate but different average-case error, such as the ACA_II_N8_Q4 and the GDA_St_N8_M4_P2. The error metrics can therefore be used to categorize circuits into more than one category. Even if the two approximate adders are equally likely to produce an error, the one with a higher average-case error will produce errors that (on average) deviate more from the exact value. This information gives insight to the designer who may value errors that are more impactful over area or power improvements, or vice versa. It is also not impossible for two structurally different adders to have the same value for both metrics, leaving the design choices up to other error metrics or area/power considerations.To discuss the effect of different design parameters on the runtime of the methodology, the CPU time is reported. Firstly, the effect of the miter size in gates/cells on the runtime is explored in The first noticeable difference between the two graphs is the jump in CPU time between the 8-bit adders and the 16-bit adders. This is due to the search space doubling with every introduced input variable, leading to an exponential growth. The other noticeable fact is that there is no pattern to the scatter graphs and hence a more influential variable affecting the CPU time. This factor can be demonstrated by looking at Figs The CPU time scales with the maximum error rate because the higher the maximum error rate the more the number of errors that need to be discovered, therefore more work.The average-case error metric has more to do with the severity of the error, i.e., the average absolute difference between the two circuits in value when considering both their outputs as integers. For example, two approximate circuits might both have one error. But one of them has the error in the most significant bit and the other in the least significant bit. Therefore, they both end up taking the same CPU time to process but different average-case error.Finally, this effect is consistent even if both benchmarks sets are considered together. However, in order to plot both benchmarks sets together, we need to consider the number of existing errors . The proposed work was effective in processing the two error metrics in one go, and provided a log of error-generating input assignments in a reasonable timeframe. It was shown experimentally that the processing time of the methodology increases, as expected, rapidly with a few inputs but only linearly with the number of errors it needs to detect.However, several challenges remain that open interesting research opportunities for future work to realize the full potential of the emergent approximate computing paradigm. It includes the study of the correlation of metrics with each other and with approximation techniques, as well as the evaluation of combined error-metrics and their impact (e.g. Hamming distance) on different applications. Another important extension of this work will be experimenting with the internal algorithm of the SAT solver and discovering if different algorithms might behave better with some circuits and worse with others. The development of parallelized ALLSAT solvers is likely to be a valuable addition in terms of speed of processing.


From the redistribution of thermal quasiparticles within the texture, we develop a model which confers a substantial effective mass on the interface, and provides a new mechanism for friction as the boundary moves. We have tested the model against existing data for the motion of an A-B interface whose motion was controlled by a magnetic field. The model allows us to make predictions for the behaviour in experimental situations which involve texture rearrangement arising from motion of the B-phase boundary.In this work, we study the influence of orbital viscosity on the evolution of the order-parameter and texture in the B phase of superfluid  Among such systems, superfluid 3He at very low temperature can condense in two phases, A and B. The bulk properties of the A and B phases are presently very well understood9. However close to boundaries, the topology of the order parameter changes and one might expect unconventional behaviour10.Many systems3He is spatially oscillated at very low temperatures , the B-phase anisotropy becomes dominated by the Zeeman splitting resulting in a large density of thermal excitations occupying states along the lB axis. Any sudden change in the spatial orientation of lB will generate substantial dissipation as the thermal excitations must redistribute.Superfluid 13, and secondly, a reactive component which can be viewed as an effective mass, which we have discussed in an earlier paper14 in which we reported a very preliminary attempt at introducing orbital dynamic ideas specifically to explain the specific case of the behavior of a moving A-B interface. Since then we have realized that our initial treatment can be extended and is not only appropriate to the moving phase interface, but can be given much more general validity allowing it to be applied to any B-phase boundary, e.g. at a solid boundary to a wire surface or moving plate. This new extension of the applicability of these ideas is the subject of the current paper.The redistribution of thermal excitations has two components. First, there is a dissipative component which can be related to the orbital viscosity, as discussed by Fisher and SuramlishvililB is the “flare-out” texture17. Here, lB lies parallel to the magnetic field in the bulk liquid far from the walls and bends to accommodate the condition that it must lie perpendicular to the side walls. The orbital axis lB also bends at an interface between the A and the B phases, stabilized by a magnetic field gradient . In the bulk B phase lB lies parallel to the magnetic field direction, but at the interface the energetically favoured orientation is that parallel to the interface18.In general, the orbital textures bend near boundaries. At solid walls, for example, say in a cylindrical geometry, with the magnetic field parallel to the cylindrical axis, the energetically favorable spatial distribution of In this work we study the effect of the change of the orbital direction at the boundaries of the B-phase in magnetic fields due to the movement of these boundaries. As we are interested in the change of the tilt angle, taking as reference the direction of the magnetic field, this can be considered a nutation motion when described using Euler angles. We apply the results to describe the dynamical behavior of an oscillating A-B interface at very low temperatures and in high magnetic fields and also to provide predictions on what will happen when the B-phase boundaries move. The dissipation will depend on the orientation of the movement of the boundary with respect to the equilibrium texture configuration.19 and Ashida and Nagai20, the energy of a quasiparticle excitation with momentum p and spin ⊥ are the energy gaps parallel and perpendicular to lB, vF is the Fermi velocity and pF is the Fermi momentum. The quasiparticle Zeeman energy, with Fermi-liquid corrections, is σ = ±1/2. Since the Zeeman splitting dominates the anisotropy, the two gap components, ⊥, can be approximated by the zero field gap Δ022. The minimum quasiparticle energy depends on the angle between the quasiparticle momentum and the lB axis. At temperatures well below Tc more or less all the excitations occupy the lowest energy states, with momenta centered around the lB axis. Thus any change in the local orientation of lB will alter the quasiparticle energies and the subsequent relaxation of the excitations back to equilibrium will proceed on a time scale τ. This provides the fundamental mechanism by which an effective mass and friction is imparted to moving boundaries.Following SchopohllB by δθ gives rise to a change in the quasiparticle energies ofθ is the angle between the magnetic field direction and the lB vector.A change in the orientation of 24μ is the orbital viscosity.This produces a viscous torqueμ was previously shown to be13τ is the quasiparticle relaxation time, N(0) is the normal density of states at the Fermi surface, and Δ we can take as approximately equal to the zero-field gap Δ0.In the B phase at low temperatures, the orbital viscosity lB always takes up its equilibrium orientation θ(r − rI) relative to the boundary interface rI. This adiabatic approximation is ultimately justified as long as the characteristic time of the motion of the boundary is much greater than τ. Any fuller treatment in the future should also take into account the dynamics of the lB texture itself. Now let us consider a small change in the position of the boundary interface δrI. This will result in the orientation of the texture in the B phase adjusting byθ is a tensor of the second order.If we assume that the B-phase orbital texture responds instantaneously to any change in the position of the interface, then we may neglect any orbital dynamics and presume that The corresponding work done on the quasiparticle distribution can be written:δrI can be written as:The corresponding effect on the boundary dynamics can be found by equating the work done by the viscous torque  to that Setting this equal to the work done on the quasiparticles, Eq. , along wγl and the imaginary part provides the effective mass ml. Note that this expression is valid for any angle between the surface normal and the angular momentum vector.Thus the real part of the orbital viscosity provides the friction coefficient lB texture is vertical, in the direction of the magnetic field B. The situation is shown in Fig. lB must turn towards the horizontal as the interface is approached18. This rotation takes place over a distance of order ξB. The corresponding texture has been calculated numerically by finding the configuration which minimizes the bending energy and the magnetic free energy simultaneously24. It is plotted in Fig. θ is the angle between lB and the vertical axis z.The expression of Eq.  allows uθ(z) is described well byx and z directions,Thus, in this particular situation the texture ϕ = 0, which corresponds to a boundary displacement parallel to the interface, the effective mass and dissipation disappears, while the maximum will occur for a displacement in the normal direction. It must be noted that the presence of any defect in the texture on the boundary or a different equilibrium configuration of the texture will change this prediction, but nevertheless we expect that there should be preferred directions for the effective mass and the dissipation.Thus, for 10. Indeed at higher interface velocities the pair-breaking has analogies with Schwinger pair-creation and the Unruh effect in particle physics25. It was found experimentally26 that the friction values of a rapid freely-advancing interface in low magnetic fields were consistent with theoretical estimates based on the Andreev scattering of thermal quasiparticle excitations27. However, in subsequent measurements in Lancaster in high magnetic fields and much lower temperatures28 the observed friction was found to be orders of magnitude greater than the theoretical predictions29. The measurements were made on an interface that was stabilized by a field gradient and which could be driven into controlled oscillation using shaped magnetic field profiles, at much lower temperatures where pair-breaking was expected to dominate the dissipation11. Furthermore, the dissipation showed non-linear behavior that appeared to depend on the frequency of the oscillations of the interface. It was pointed out that the motion of an oscillating A-B interface in high magnetic fields and at low temperatures should be dominated by the orbital viscosity and by a significant effective mass arising from the redistribution of thermal quasiparticle excitations confined in the B-phase order parameter texture14. The change of the surrounding texture and energies of thermal quasiparticles should contribute to the effective mass of the interface and also produce dissipation arising from the orbital viscosity13.We can apply the model to an oscillating A-B phase interface. The dynamics of such an interface is particularly interesting at low temperatures. It has been proposed that the interface has an effective mass and there is dissipation due to pair-breaking and Andreev scattering when it moves27nAB the surface normal directed toward the B-phase, and χAB being the difference in the magnetic susceptibilities of the A and B phases.The motion of the interface can be described byξB over which lB changes direction is smaller than the size of the boundary and the effect of the side walls in the form of a meniscus can be neglected. We note, that in the A-phase the preferred orientations of the orbital vector lA in the bulk and on the surface are the same, so the orbitropic effect does not manifest in the A-phase when the interface is moving along the direction of the magnetic field.To simplify the calculations, we will again assume that the B-phase response to the changing position of the interface is instantaneous is and that the boundary remains flat during its motion. This last approximation is valid as long as the healing length dV = Adz with A the area of the interface, readsUnder those conditions Eq. , after sθ changes exponentially from π/2 to zero over the textural healing length ξB, as given by Eq. 17) in Eqm = ml and γ = γ0 + γl. The contribution γ0 comes from Andreev scattering and pair breaking29, or any dynamical friction with the walls.So that the friction and mass parameters of Eq.  become m11 and in more detail by Arrayás et al.14. Briefly, the measurements were performed on a sample of superfluid 3He contained in a sapphire tube, closed at one end and at the other end connected through an orifice to the inner cell of a Lancaster-style nuclear cooling stage. The volume inside the tube acts as a quasiparticle black-body radiator. Any dissipation in the superfluid contained in the tube generates an excess of quasiparticle excitations which are lost through the orifice to the bulk superfluid outside. From the temperature (≡ quasiparticle density) measured inside and outside the radiator volume we can deduce the energy flux flowing out of the orifice, which provides a very accurate measure of the dissipation in the radiator volume. (These devices are remarkable sensitive being able to detect energy inputs on the 10−16 W level.) The sapphire tube has internal diameter 4.3 mm and length 44 mm. The experiments were carried out at a pressure of 0 bar. A solenoid stack was used to create a shaped magnetic field profile to stabilize the A-phase in the bottom of the tube in fields above the transition field Bc = 340 mT31, whilst maintaining the top of the tube in the low field B-phase. Once the A-B interface was established across the tube, a small additional alternating field was applied to oscillate the interface vertically over a range of frequencies.Let us now test the results on real experimental data, those plotted in Fig. γl and ml for the experimental data in Fig. 22; Δ0 = 1.76kBTc with Tc = 0.929 mK at 0 bar pressure, from32; N(0) = 1.0 × 1051 J−1m−3, from33; and T = 155 μK . We assume ξB = 0.1 mm, using some estimations for fields close to Bc35.To evaluate τ and the frequency independent dissipation term γ0, which is an additive term to γl in . For the set of experimental data in the case of the 214μT drive, we have also considered the fitting with no extra constant dissipation. The lack of experimental data at higher frequencies prevents us from drawing further conclusions. We remark that texture bending could be a function of the driving amplitude and remains to be further investigated.The only unknown parameters in our model are the quasiparticle relaxation time to γl in . The mod14. To model the double-plateau profile of the data as a driven damped harmonic oscillator, the effective mass and the dissipative coefficient must be frequency dependent, as shown by Eqs 14. To moWe should emphasise that the double-plateau profile is certainly real and not associated with any saturation effect from heating by the moving boundary. In other similar measurements we certainly see no temperature gradients in the cells which would imply heating at this point.11 we followed the then current wisdom that these effects were the result of Andreev reflection, but a simple linear friction fit on that basis simply did not reproduce the observed data. In ref.14 we had understood that the effect arose from orbital precession of some sort and managed to achieve a more realistic fit, albeit for a very restricted version of the interface. The current model now reproduces the features of the system rather well and is completely general for all configurations of the interface. When the original measurements were made, the high dissipation turned out to be a nuisance as we had no idea what we were actually seeing. Now that we have a clearer understanding of the contributing processes, this would be a good time to revisit the experiment but now with a clearer view of the contribution of the orbital evolution. In future experiments the orbitropic effect can also be studied in other, less complicated situations. For example, in a magnetic field in the vicinity of a thin superconducting wire carrying alternating current. The magnetic field generated by this current will reorient the texture in 3He-B, thus causing dissipation (the magnetic field on the surface of a 0.1 mm thick wire carrying 1A is 4 mT and the field gradient is 80 T/m).The data are of great interest, and especially because the dissipation is so much higher than would be naively expected without taking into account the response of the orbital moment which we model here, which can be seen from the way the ideas of how to deal with the data have evolved. In ref.37. The paper predicts a very peculiar form of a magnetic wave that can be excited at the interface: as the two superfluid phases have different magnetic susceptibilities, the phase transition between them is accompanied by a change in magnetisation. The author estimates that the effective inertia of such a wave is not large enough for the wave to be excited in a typical experiment. However, the orbitropic effect will endow the wave with a high value of the effective mass: at the resonant frequency of the wave  our calculation (ml = 8.5 × 10−7 kg/m2 which is about 100 times greater than that predicted in37. As a result, the dissipation rate of 35 s−1 is small compared to 2πfres and makes the observation of the magnetic phase wave more realistic.We would also like to comment on the results of a recent paper concerning the motion of the A-B interfaceculation  gives mllB. We have estimated the effective mass and dissipation associated with this motion, assuming adiabatic approximation of the influence of the orbital dynamics on the boundary dynamics. The main result is expressed in Eq. (In conclusion, in this work we have considered a new dissipation mechanism based on the redistribution of thermal quasiparticles in the orbital texture. We have investigated its effect at the boundaries of the B-phase, and studied the consequences of the change of the nutation angle of the orbital vector d in Eq. . Further


Integrated assessment models (IAMs) of climate and the economy provide estimates of the social cost of carbon and inform climate policy. With the Nested Inequalities Climate Economy model (NICE)  contains supplementary material, which is available to authorized users. Models of optimal economic responses to climate change are contingent on highly uncertain estimates, including about the future societies with which climate will interact. Nevertheless, incomplete estimates exist, and when put into integrated assessment models (IAMs), they serve to provide some guidance on the extent of effort society should dedicate to this problem Stern , 2009.There is no consensus that a high mitigation effort is warranted, but there are several instances of such recommendations by researchers using these models—most notably the Stern Review, which computes a high social cost of carbon based on very low discount rates Stern . Other rIn this paper, we investigate the interactions between several of these individually important factors by providing a systematic study of the comparative importance for optimal mitigation of inequality aversion, time preference, catastrophic damages and intragenerational distribution of damages and abatement costs. While a longer list of factors could be drawn, this selection enables us to make two main points. First, the direction by which inequality aversion influences optimal mitigation depends on the distribution of mitigation cost and damages within countries and the time preference parameter. Second, the distribution of mitigation costs and damages is of prime importance for optimal mitigation, whereas catastrophes become relevant only when inequalities are a low concern. Our investigation of these issues is made possible by further development of the Nested Inequalities Climate Economy (NICE) model  generation, as implied by the Ramsey equation. At intermediate levels of discounting, these two effects balance out, and inequality aversion has a small effect on optimal carbon prices.Our first point about the importance of the distribution of damages and mitigation cost relates to Thomas Schelling’s conjecture that properly accounting for inequalities within generations could lead the inequality aversion parameter to have an effect opposite to what is suggested by the Ramsey equation . These parameters are grounded by some authors on ethical considerations and by others on observed savings rates coupled with very strong assumptions about savings behaviour. But the common result suggested by the Ramsey equation is that increasing either one of these parameters delays mitigation. That is because the Ramsey equation states that the discount rate on costs and benefits (as measured in dollars) is given byg is the average growth rate of consumption between now and the discounted period. This equation suggests that a greater value for either ρ or η will raise the discount rate on consumption and therefore delay mitigation, given the positive growth rate assumed by these models.More generally, the growth rate assumed in leading IAMs generates substantial inequalities between generations, as the assumed growth rates and damage functions imply that future generations will grow richer and richer even under business as usual .i, j, and t are the region, quintile and time indices respectively. It is clear from this specification that raising inequality aversion leads to a lower valuation of damages and mitigation costs when they fall on richer groups of individuals than if they fall on poorer groups, whether they are separated in time or contemporaneous.A more detailed description of the model is provided in the Most of the equations of motion are inherited from RICE. The main difference is that in NICE, regional consumption is split into five quintile shares. This is done by aggregating country income distribution data for all countries in a region  i iξ, for ξ and ω, consider a ‘regional economy’ comprised of two  consumption groups A and B, with A consuming USD4000 and B USD40000 a year. If this region suffers 5% damage from climate change, they jointly lose USD2200. If ξ = 1, A loses 200 and B loses 2000. If ξ = 0, both A and B lose 1100. If ξ = − 1, A loses 2000 and B loses 200. B goes from losing 5 to 2.75 to 0.5%, while A goes from losing 5 to 27.5 to 50% of pre-damage consumption. If the same region experiences a 2.5% abatement cost from climate change mitigation activities, then that total joint cost to groups A and B is USD1100. If ω = 0, both A and B pay 550. If ω = 1, A pays 100 and B pays 1000. If ω = 2, A pays 10.9 and B pays 1089.1. A goes from paying 13.75 to 2.5 to 0.3%, and B goes from paying 1.38 to 2.5 to 2.7% of pre-mitigation cost consumption. In this way, the values of ξ and ω affect only the distribution of damage and mitigation cost within a region, and not the total amount of regional damage and regional mitigation cost.To further illustrate the meaning of ξ, depends on where and how the climate changes and modifies the ecosystem at a sub-regional level, on how vulnerable the populations are given the organisation of the economy and the infrastructure set-up, and on policy response.ξ has not received much scrutiny so far in the empirical literature, perhaps partly due to the fact that the importance of this parameter has not previously been demonstrated. However, many studies argue that the poor will disproportionately suffer from climate change . We consider that a relevant range for ξ in the present investigation is from − 1 to + 1.The distribution of damages, and thus the value of ω, is even more dependent on policy decisions. Several studies , the optimal price is the same across panels and high ($233/tC in 2015) relative to commonly computed values with IAMs  that this is true for a range of values of η and equally true for the distribution of mitigation cost. The greater η, the more important it is to get the correct value of the elasticity parameters ξ and ω.At zero inequality aversion  or when the distribution of damage is more progressive (bottom row). In the first case, less mitigation strongly benefits the current generations’ poorest, so more concern for the poor reduces optimal mitigation. In the second case, more mitigation benefits the future rich more than the poor, so more inequality aversion again favours the current poor inducing less mitigation.The opposite is true in the rightmost column. Here, the future rather than the present poor dominate the evaluation, and the Ramsey equation is upended, as Schelling conjectured might happen. In the top row, mitigation costs are borne largely by the current rich. This shields the current poor from increases in mitigation, making greater inequality aversion result in higher optimal prices.ρ = 0%, for example, we find that the Schelling Reversal does not take place: all panels have optimal prices that decrease with η. This is because of a level effect. At such a low discount rate, the optimal prices are already so high that the most important damages are avoided. The optimum is already skewed in favour of the future, so increasing inequality aversion is more sensitive to the present poor, irrespective of the distribution of costs and damages. When ρ = 1%, we have an intermediate case where the effects balance out and the optimal price is not particularly sensitive to η.Under the central assumption in the middle panel, the balance favours greater carbon prices at higher inequality aversion in the future, but an opposite  effect for present prices. This highlights that there is also a dynamic component to the interaction between the distribution of costs and benefits and inequality aversion. Relative to the left column, the price paths become steeper with greater inequality aversion. Under those assumptions, greater inequality aversion results in a present reduction in mitigation effort because it improves the situation of the current poor. However, as economic growth increases the living standard of the poorest in the near future, the optimal price increases rapidly if inequality aversion is high. In the supplemental material, we produce companion figures to Fig. Altogether, these results reveal the supreme importance of the distribution in income, damages and mitigation costs in the determination of the optimal carbon price. They confirm that Schelling’s Reversal can occur and that this depends in an intuitive way on the distribution of damages and mitigation costs within regions: the more the poor stand to benefit from mitigation and the less they pay for it, the more likely the reversal. However, this also depends on the rate of pure time preference, which must be sufficiently high (and imply a sufficiently modest mitigation) for the reversal to be observed.ξ) the way in which that total amount of damage is distributed across the income groups of every region. Here, we consider two additional damage specifications in which there is significantly greater overall damage once temperatures increase beyond what has been observed in the historic record. As described at the end of Sect. So far, we have taken the total dollar amount of economic damage resulting from climate change as given by the original quadratic damage specification of the RICE model, and only modulated . As described at the end of Sect. In Fig. In the left panel, the elasticity of damage is low, and therefore, the optimal carbon prices are high. Comparing the different damage specifications, we see that all three are almost identical. If, however, the elasticity of damage is high (right panel), the optimal prices and mitigation efforts are smaller, and the difference across damage specifications becomes more significant.The results in Fig. We note that while this analysis holds for optimal carbon prices, the economic damage that results from having a non-optimal policy (i.e. a policy that is too lax) may be quite large and significantly affected by the alternative assumptions about damage considered here. Incorporating uncertainty over the damage function and other key parameters (including climate sensitivity and other parameters not even discussed here) might also lead to a stronger policy response.ρ, η, and ω. The overall pattern is similar to the one in Fig. ρ and low values of ω). The effect of η cannot be summarised in such a way because of the ambiguous effect of η on the optimal policy.In the supplemental material, we have included plots corresponding to the three panels of Fig. ρ and η), but agreed to some extent that high levels of effort are warranted if uncertainty is taken into account, with a low probability of very strong damages on aggregate output. Qualitatively, this effect can be seen in the right panel of Fig. distribution assumptions are similar to what Weitzman had in mind, and there is a clear effect.Weitzman  disagreeξ and ω) with the variation due to the different aggregate damage specifications, it becomes apparent that the way in which costs and benefits are distributed across income groups has a much greater impact on the optimal carbon price than even the extreme damage specification by Dietz-Stern. Only if η = 0 do distributional considerations become less important that the possible aggregate catastrophe, and no example in the IAM literature considers that possibility. This is an important lesson. By neglecting the intragenerational distribution of inequalities (within regions and countries), the literature had to look at global catastrophes in order to explore potential reasons for strong mitigation. With inequalities and a skewed damage distribution, the social catastrophe looms large even at less extreme temperature increases.However, when optimal mitigation effort is already high, as is the case with a highly regressive damage distribution, the inclusion of an aggregate catastrophe in the damage function has no effect on the optimum. Furthermore, comparing the variation in optimal carbon prices for the range of values taken by the distributional parameters  ways on several conditions. As Schelling argued, the distribution of damages and abatement costs is the essential ingredient in the analysis and involves both climate, economic and political assumptions.Using NICE, we have investigated the interactions between a number of factors that can, in isolation, have an important effect on the strength of optimal mitigation—namely, pure time preference, inequality aversion, inequalities in the distribution of both damage and mitigation cost between rich and poor and a damage function that does or does not assume catastrophic impacts at high global mean temperatures increases. We have shown the comparative importance of these factors by displaying optimal carbon price trajectories that arise from the wide variety of combinations that are possible given the primary range of disagreement over each factor.To frame the discussion, we began by articulating Schelling’s conjecture that properly accounting for inequalities could lead the inequality aversion parameter to have an effect opposite to what is suggested by the Ramsey equation. We showed that the Schelling Reversal does indeed happen but only when a set of conditions are all satisfied: roughly, when climate damages disproportionately harm the poor, when mitigation costs are not disproportionately paid by the poor, and when discounting is sufficiently high making effort at the optimum low; in these circumstances, increasing inequality aversion leads to faster optimal mitigation. Otherwise, intergenerational inequalities tend to dominate. The Schelling Reversal emerges especially strongly in situations of a progressive distribution of abatement costs and a regressive distribution of climate damages.We also showed that for most values taken by the other parameters, the effect on optimal policy of adding catastrophic damages with certainty at 4 or 6 °C above pre-industrial temperatures typically has a smaller effect than the other parameters investigated here, especially when compared to the effect of sub-regional inequalities. The addition of potentially catastrophic damages has a larger effect on optimal policy when damages are proportional to income than when they fall more on the poor, because the latter requires strong mitigation for reasons independent of the potential for catastrophic damages, which prevents temperature from increasing to the point where the elevated damage function would be relevant. However, we note that our experiments are deterministic and do not explore uncertainty as well as the economic consequences of non-optimal carbon policies, both of which are important further issues, and which we explore in work in progress.In sum, NICE provides answers to a number of pressing questions about the comparative importance of familiar factors such as discounting and higher order damage terms for the determination of the price of carbon. This paper has especially highlighted that distributional catastrophes may loom larger than global catastrophes, although inequality aversion tends to shift the focus towards the future poor rather than the present poor only when time preference is high, i.e. when mitigation is low.ESM 1(PDF 1285 kb)


Road traffic accidents are the major cause of mortality among people aged 15–29 years in Africa. World Health Organisation (WHO) and the World Bank launched a Decade of Action for Road Safety in 2011 with the goal of halving the number of injuries and deaths on the roads. No progress has been reported in Low and Middle Income Countries (LMICs) and the number of deaths remains very high. To reach the target set, there is a need for interventions in several areas. This scoping review proposes to produce a synthesis by identifying the kinds of interventions and outcomes which have been carried out on the African continent. Using the scoping studies method, 23 articles were selected and analysed. The study shows that interventions were developed in four fields: road safety policy, health education, safety equipment and data collection. It shows also that there were records of interventions in only twelve countries, mostly in Eastern and Southern Africa. The main conclusion of this study reveals both a lack of road safety interventions and shortcomings in the assessment of those performed and selected for our study. Road Traffic Injuries (RTIs) are the eighth cause of death in the world and there has been a 46% increase since the 1990s . Added tThe highest number of deaths occurs on the African continent, i.e. a rate of 26.6 deaths per 100 000 inhabitants . FurtherThe Decade of Action for Road Safety 2011–2020, developed by WHO and the World Bank, provides an overall framework for road safety activities  based onAlthough the Decade of Action for Road Safety has been underway for 8 years, the number of deaths has remained high and there has been no significant improvement. Half-way through the decade, mortality rates have stagnated worldwide. The number of fatalities is still high in LMICs, and is increasing each year . In ordeTo reach this 3.6 target, and looking beyond interventions on infrastructures, the effectiveness of which is understood, interventions must be conducted in the fields of awareness-raising, provision of rescue services and also public policies and regulations which contribute to the reduction in accidents and their impact on health. Therefore, the purpose of this article is to produce a state of knowledge on the interventions that have been implemented on the African continent.The objective of this article is to provide a state of knowledge on interventions to reduce road accidents and injuries in Africa. This paper describes the interventions, their results and evaluation methods. Our synthesis complements two recently published systematic reviews on the topic of Road Traffic Injuries in Africa. Staton’s review , publishThe principal goal of this review is to take stock of the knowledge of interventions aiming to reduce the number of accidents and injuries. By intervention, we mean “a coherent, organised and structured set of objectives, activities, means and people who conduct it, implemented in order to transform a problematic situation.” . The seaThe methodology of research employed is based on the one developed by Arskey and O’Malley . It compThe research question endeavours to identify, take stock of, describe and analyse all the interventions that aim to reduce road traffic accidents and injuries in Africa. We have chosen to include all publications known since the 1950s in order to identify all types of interventions that have been carried out in Africa so as to enumerate them and highlight successful experiences. To address this question, the two principal eligibility criteria for selecting the papers for this scoping review concern i) the geographic area (countries on the African continent), ii) the fields of interventions . Implicitly, the selected interventions do not target specific ages, specific methods or geographical environments. The selected interventions must be detailed enough to complete the Tidier grid . All priSeveral databases were used : ScienceDatabase queries were adapted to each database, in accordance with the appropriate request vocabulary. For certain bases, the performance of query criteria (Booleans) is more important. This is the case, for example, for MeSH terms in the MEDLINE and CENTRAL databases .Additionally, a search into grey literature was conducted as well as a manual search from article references. The OpenGrey, National Institute for Health and Clinical Excellence (NICE), National Guideline Clearinghouse, Social Care Online and The Grey Literature Report databases were therefore used to identify the literature. Lastly, a final analysis was conducted in Google Scholar to identify papers not listed in the aforementioned databases.The search strategy focused on references in French and English as described below :("accident traffic" OR "road traffic injury" OR "Road safety" OR "Road crash" OR "road accident")AND                             AND                             To optimise the query, the names of all African countries and their regional names were added to the search strategy as well as the term “africa” so as to maximise the selections coming from titles and keywords. The search strategy was also given in French. We sought the expertise of a librarian for the definition of the search strategy.The selection of studies was done in four stages representing four triage phases of the references selected: general identification, screening, eligibility and lastly selection of the articles included.The first two authors completed the selection of the publications. They assessed the titles and abstracts independently. The selection was then compared to retain the list of publications included. Any rare cases of disagreement (n = 8) were dealt with by consensus. The reasons for exclusion of the articles were mainly associated with a geographical area of intervention outside Africa; the description of legislation or public policies not implemented in the field; and theoretical descriptions of interventions that could be carried out to reduce trauma but not applied. Most of the excluded articles included in their titles or summaries a reference to interventions, but they were not carried out. Details of the selection can be found in the presentation of the PRISMA flowchart  and PRISThe analysis grid used was taken from the template for intervention description and replication (TIDieR) checklist (Appendix 2) . It is bThe search allowed us to identify 241 articles. They were compiled in the Zotero bibliographic software tool which made it possible to sort through the references and remove duplicates. Once the four triage phases had been completed, 23 articles had been retained. Grey literature documents (n = 55) were not included in the final selection as they did not deal exclusively with detailed interventions, but with one-off actions included in an overall approach to road safety. Most of the publications focused on the last 20 years, with an increase since 2014  2. NigerMapping the distribution  of publiWe analysed the interventions using an analysis grid with 17 criteria according to the Tidier grid. Analysis of the publications identified four principal fields of intervention . The mosFive types of intervention were distinguished and were unevenly distributed according to their fields. The media, for example, covered five of the eight interventions in the field of health education. Other interventions, whose overall objective was a reduction in accidents, were actually a combination of several types of intervention. In Côte d’Ivoire, a public policy proposed 10 types of intervention  at the sOf the 23 articles, eight focused on the assessment of media activities in road safety. Three articles pointed to campaigns implemented in the development of vast road safety programmes in Kenya as early as 1980 , NamibiaThe goal of communication campaigns is to warn and raise awareness in connection with a given problem. Some campaigns provided information on dangerous driving –24 or taIn Ghana , televisBetween 2011 and 2013, researchers , 26] te te26] teIn Zambia , a preveThree references related to training sessions. Two of them described a single intervention conducted on motorcyclists in Nigeria, at Uyo , 28]. T. T28]. TIn Rwanda, in 2011, a training course was set up with the purpose of improving coverage of first aid . These cOther interventions described in the part dedicated to policy and programme safety used training sessions in addition to another intervention –32Two publications dealt with interventions using new technologies to analyse road traffic accidents and improve road safety. These two studies concerned Nigeria.The first one  proposedThe second study  set out A single article presented an intervention in a real-life situation with the implementation of a data collection system using smartphones in Nigeria . The secArticles with regard to implementing interventions in the legislative field did not form part of our scoping review. The selection dealt mainly with public policies in which laws were passed and the impact analysed. These were only general analyses on several countries, in Africa, and rarely on precise cases. The Anderson et al. article , for exaThe article by Abegaz et al  assessedThe article by Bishai  presenteThree articles dealt with the distribution of safety equipment in Tanzania and the world , 38], , . Three oIn 2008, Liu et al  conducteAnother intervention in rural Tanzania organised the distribution of road safety equipment to different categories of the population . Before The article by Obionu in 1985  dealt wiLastly, a final study analysed the calming measures installed since 2011 in Durban, South Africa . These dOur scoping review on the description of interventions to reduce the consequences of road accidents in Africa confirms the low number of interventions on the continent. It was also noted that the implementation of interventions is relatively recent as 13 of the 23 articles have been published since 2011. They correspond in part to the impact of the Decade of Action for Road Safety (2011–2020) and focus on the five pillars. This scoping review is a significant complement to the systematic review published in 2016 by Adeloye. The latter retained only three interventions in Africa out of the 18 chosen for their publication dedicated to LMICs. The same is true regarding the low number of interventions, despite numerous calls for papers published in scientific journals –43, and Without accurate data, it is impossible to develop effective road safety interventions, and it is impossible to evaluate the effectiveness of those interventions that are in place. In most sub-Saharan African countries, researchers cannot rely on the data available from government sources to accurately quantify the problem, measure the impact of injury prevention strategies, or appreciate how the patterns of injury evolve over time.”The geographic analysis of the interventions identified in this scoping review revealed a concentration of intervention research in Anglophone Africa supported in part by the Bloomberg Foundation and the Road Safety Programme. Francophone Africa, in the west or centre of the continent, presented very few interventions. Besides Benin, Francophone Africa was not represented despite mortality rates that were amongst the highest in the last WHO assessment . Could iThis scoping review also reveals that several interventions perform combined activities ,20], il, il20], Considering the assessment methods, the greatest proportion provided meta-analyses from which it was difficult to extract the initial data and therefore to reproduce in another context. In the end, nine articles focused on effectiveness. The authors tended to validate the outcomes of their interventions without necessarily supporting them by robust assessments which were themselves difficult to carry out, given that there was no monitoring of accident and victim numbers, or resulted from an estimation. We made the choice of including them in this scoping review, making it clear in the synthetic table the type of assessment and demonstration of their effectiveness or otherwise. The existence of these interventions and their implementation remain relevant, useful and rare in Africa. These experiments will serve to build other intervention projects which should be adapted to the countries’ contexts.All the fields of prevention and types of intervention had something to do with awareness raising, in particular via the media or communication. However, in the examples used in this scoping review, it can be noted that the media or communication tools involved were adapted to the contexts of the countries in which they were applied, like the stickers in Kenya, or raising awareness via community radio stations in Zambia. If these actions appeared effective, they remain difficult to assess and generalise. The second field which seems to be the most effective relates to basic first aid training for nurses and emergency doctors when treating casualties. Lastly, the third field is traffic enforcement with a dissuasive effect through the presence of security forces and checks. This combination of interventions is reminiscent of Haddon’s conceptual framework  where, aThe success of interventions on traffic enforcement must be associated with the context of the countries, given that differences exist in the effectiveness of the administrative and legal systems within countries on the African continent. Populations’ level of resources is also crucial in the use of fines. The impact would be negligible, would indeed worsen living conditions, if applied to the most disadvantaged populations. Lastly, road safety education as well as raising traffic protection awareness seem to be the interventions which are most adapted to African countries even if a strategy for changing behaviours must be adopted, appropriate to each socio-economic and national cultural context. The overall analysis of this scoping review reveals that there is a lack of interventions in the field of road safety on the continent, even though the World Decade for Road Safety and many awareness raising activities by the scientific communities had stressed the urgency of the situation. Available funding focusing specifically on the subject is certainly insufficient to produce new knowledge emanating from research. A final point to take into account is that the few interventions conducted could be better assessed so that elements of replication could be obtained in order for the experiments to be directed to other places and on other scales.S1 Appendix(PDF)Click here for additional data file.S2 Appendix(PDF)Click here for additional data file.S3 Appendix(XLSX)Click here for additional data file.


At least one-third of all people with epilepsy have seizures that remain poorly controlled despite an increasing number of available anti-epileptic drugs (AEDs). Often, there is an initial good response to a newly introduced AED, which may last up to months, eventually followed by the return of seizures thought to be due to the development of tolerance. We introduce a framework within which the interplay between AED response and brain networks can be explored to understand the development of tolerance. We use a computer model for seizure generation in the context of dynamic networks, which allows us to generate an ‘in silico’ electroencephalogram (EEG). This allows us to study the effect of changes in excitability network structure and intrinsic model properties on the overall seizure likelihood. Within this framework, tolerance to AEDs – return of seizure-like activity – may occur in 3 different scenarios: 1) the efficacy of the drug diminishes while the brain network remains relatively constant; 2) the efficacy of the drug remains constant, but connections between brain regions change; 3) the efficacy of the drug remains constant, but the intrinsic excitability within brain regions varies dynamically. We argue that these latter scenarios may contribute to a deeper understanding of how drug resistance to AEDs may occur. One of the many challenges for neurologists treating epilepsy is that at least one-third of all patients have seizures that remain poorly controlled despite the increasing number of anti-epileptic drugs (AEDs) now available, so-called drug-resistant epilepsy (DRE). This number has remained frustratingly static, with no significant improvement over the past 30 years In recent years, our understanding of the role that the topology of large-scale brain networks plays in shaping neuropathology has greatly advanced 2A dynamic network model was used to simulate networks of six interconnected brain regions with asymmetric directed connections between them. The model is constructed as follows. Our starting point is an equation previously introduced in the context of epilepsy z, whose output may be thought of as a proxy for the electrographic activity recorded within a single channel of EEG overlying that brain region. This equation was chosen as it allows two different types of dynamic behavior to simultaneously coexist. One is a low-amplitude steady-state solution . The other is a high-amplitude oscillatory state  This is a stochastic differential equation and is used to describe the dynamics within a single brain region. Such an equation describes the evolution over time of a variable ω determines the frequency of oscillation. An appropriate choice ensures that the output oscillates around three cycles per second (similar to the frequency seen during seizure activity). dw is a noise term that has an amplitude defined by α. Depending on the size of the perturbation delivered by the noise term at each instance in time, it is possible to transition from one dynamic state  to the other , see The parameter λ effectively controls the likelihood of transitioning between the interictal and ictal states within a brain region, effectively how ‘excitable’ each brain region is, and is defined by the following ordinary differential equation:The variable λbaseline, which is inhibited by the overall level of excitability λ, the activity in the system | z |, and the effect of the AED (λAED). A system with low baseline level of excitability (λbaseline close to 0) can be thought of as a ‘normal’ brain region whereas a region with λbaseline close to 1 can be thought of as strongly ictogenic. Note that the effect of the AED is modeled within the range λAED ∈ , which effectively reduces the excitability of a region, making the transition to ictal activity less likely. In this representation, we do not ascribe any neurobiological mechanism to the reduction in excitability elicited by the AED, rather, we think phenomenologically of the desired effect of an AED being to reduce the level of emergent ‘excitability’ within a brain region. As shown in λbaseline and α. If λbaseline is close to 0, only a large value for α would drive the system towards the ictal state whereas if λbaseline is close to 1, even small values of α would be enough to drive the system into a seizure. τ is a given time constant, which effectively determines the duration of a seizure. Typical values of parameters and variables are presented in The overall transition likelihood depends on a fixed parameter for the system It is important to reiterate that this is a purely phenomenological description that aims to describe the theoretical phenomenon of a transition into the seizure state; this means that a model simulation replicates at a conceptual level the fundamental components of the transition into a seizure but does not have the same richness and level of detail as an actual recording of a seizure. This is much in the way that an animal model typically presents a phenotype that may be related to the human condition but is, of course, not a substitute thereof.A – effectively a set of ‘0’s and ‘1’s where ‘1’ indicates a connection between two regions), with the dynamics of region i described by:We can extend these equations for a single region into a network of coupled regions, where the dynamics of each region would correspond to the activity captured by an EEG channel overlying that region. The consequent “dynamic network model” consists of coupling together N such regions (whose topology or structure is defined by the adjacency matrix f(zi) is defined as previously. In the context of this phenomenological description, a connection between two regions is “synchronizing”. This means that region i will influence region j to exhibit the same dynamic state that it is in  if there is a connection between regions i and j (due to the term (zi − zj)). As a conceptual study, we do not attempt to ascribe any underlying neurobiological context  to these connections; defined by the matrix A, they represent an abstract representation of a brain and are not derived from imaging data , or functional magnetic resonance imaging (fMRI)). Herein, we consider a set of six interconnected brain regions as a pragmatic balance between complexity and ease of illustration.Here, β, the global coupling strength, which determines the overall influence that connected regions within the network can have on each other. Whether a transition to seizure is observed is therefore dependent on the interplay between the level of excitability and the intrinsic noise term within each brain region and the synchronizing effect of other brain regions as a result of the given connections within the network.A critical determinant of level of influence is 3The starting point for each scenario was a six-region network see , where t3.1The first scenario we consider is designed to illustrate the classical concept of the ‘honeymoon’ period A. In thi3.2In the second scenario, we consider the role that neural plasticity, as reflected in the evolution of large-scale network structures between brain regions, may play B. Again,3.3In the final scenario, we consider the role that neural plasticity, as reflected in a change in internal dynamics within a brain region, may play C. In thi4Understanding the mechanisms of drug resistance is a necessary step to develop strategies to mitigate this problem Within this framework, we assume that the action of an AED at the macroscopic level is to reduce the level of excitability within brain regions. We present three scenarios by which tolerance to AEDs can occur. The first assumes that a brain network remains relatively static over time while the efficacy of the drug diminishes. This scenario can be thought of as the classical case: drug efficacy diminishes because of some unknown  mechanism, and increased excitability of the brain leads to the return of recurrent seizures. However, we further describe two other scenarios where seizures can recur. In both cases, there is no change to the efficacy of the drug, rather, properties of the large-scale brain network evolve dynamically over time. One case explores alterations to connections between brain regions, as we might expect from neuroplasticity Such future studies focused on understanding how functional representations of brain networks (either from EEG or fMRI) vary over time may play a critical role in improving our clinical understanding of the relative contribution of microscopic and macroscopic variables in seizure generation and therefore the overall response to AEDs. For example, long-term EEG monitoring could enable the identification of changes in network structure or model properties related to changes in medication or dosage. To date, recordings of sufficient duration to capture changes of this nature have not been available to study changes in the dynamics of brain activity over the long periods necessary but will likely soon be possible, for example using subscalp electrodes Medical Research Council (via MR/N01524X/1). John R. Terry and Mark Cook received financial support from a Royal Society International Exchanges Award (grant number IE170112). John R. Terry further acknowledges the generous support of the Wellcome Trust Institutional Strategic Support Award (WT105618MA), and the EPSRC (via Centre Grant EP/N014391/1). We have no conflict of interest to declare. We confirm that we have read the Journal's position on issues involved in ethical publication and affirm that this report is consistent with those guidelines. Full Matlab scripts will be made available on an appropriate repository .Wessel Woldman received financial support from the