A two-dimensional self-consistent fluid model was employed to investigate the spatiotemporal nonlinear behavior in an argon glow-like/Townsend-like dielectric-barrier discharge (DBD) at atmospheric pressure. The discharge is characterized by a major current pulse with a residual one ahead per half cycle of the external voltage. The two current pulses are operated in glow mode, but with Townsend mode between them. Contrasting spatial discharge structures are complementarily presented not only at two current pulses in the same half cycle but also during the discharge in the two adjacent-half cycles, resulting in the formation of a unique nested complementary pattern each cycle. This peculiar behavior mainly lies in the fact that sufficient charged particles are trapped in the gas gap due to the last discharge and able to dominate the subsequent discharge through the “spatial memory effect”. The charge transport regime reveals that this nested complementary pattern is presented only in a limited range of driving frequency. The self-organized patterns have attracted much attention due to their complexity and diversity, which are commonly presented in many systems of different origins such as chemical34567910412In this paper, we report the first simulation results that unambiguously demonstrate that there are contrasting spatial discharge structures complementarily presented not only at two adjacent current pulses in the same half cycle but also during the discharge in the two adjacent-half cycles in an argon dielectric-barrier discharge. A nested complementary pattern is observed. This discharge is characterized by a major current pulse with a residual one ahead. The spatial discharge structure in the negative half cycle is just opposite to that in the positive half cycle, although the current pulse is symmetric in the two adjacent-half cycles. This behavior is closely associated with the residual discharge that has been identified both in simulations and in experiments1415r−z), as shown in t4 = 350.7 μs) and the weakest at the negative major one (t9 = 375.6 μs), while it presents the weakest at the positive residual current peak (t2 = 344.3 μs) and the strongest at the negative residual one (t7 = 369.4 μs). Compared with the temporal evolution of current density at r = 2.8 mm over one period, the case at the center point (r = 0 mm) is reversed at these four current peaks. The temporal evolutions of current density at these radial positions indicate that the current density appears to be asymmetric in a local discharge region and has different waveforms in the positive and negative half cycles. This indicates that the temporal nonlinearity exists in the discharge. In addition, the distinct current density at different radial positions at a fixed time indicates that the discharge intensity is non-uniformly distributed along the radial direction and the discharge exhibits spatial nonlinearity. These discharge characteristics suggest that temporal and spatial nonlinearities simultaneously exist in the argon DBD. This temporal evolution of current density is quite different from those observed both in simulations and experiments performed by Wang et al.et al.Our simulation was carried out based on a two-dimensional (2D) self-consistent fluid model in an axisymmetric cylindrical coordinate (t2) in the positive half cycle, as shown in t4), the minimum of electron density appears in the center, and the density increases along the radial direction and reaches its maximum in the periphery of the gas gap, showing a periphery-advantage distribution. It is known that the discharge current density, whose spatial distribution can be visualized by the discharge luminance in experiment, produces charged particles in the gas gap12t7) presents a periphery-advantage distribution, which is similar to the case at the positive major current peak, but contrary to the situation at the positive residual current peak. Subsequently, the electron density manifests itself as a center-advantage distribution at the negative major current peak (t9), as indicated in t1−t10 marked in To demonstrate the formation of nested complementary pattern, the spatial distributions of electric field and electron density are plotted in t1 = 341.6 μs) before the positive residual current pulse, more charges assemble in the center of the gas gap due to the “spatial memory effect”, as can be seen in t1 in t2 = 344.3 μs in t2 in Since space charges trapped in the gas gap are remained in the last discharge and they have no enough time to diffuse and transfer to other places under the low electric field in a very short phase, the similar spatial distribution of charges is maintained as that in the last discharge. This behavior is defined as “spatial memory effect” or “footprints” in the 3-dimentional discharge space, instead of the memory effect on the 2-dimentional dielectric layer for surface charges to be deposited ont3 = 346.3 μs) after the positive residual current pulse, most of the space charges are exhausted and few new charges are generated in the gas gap due to the low electric field  give rise to the reverse of spatial discharge structure between the two adjacent current pulses in the same half cycle. The remain of large amounts of space charges in the gas gap (after the major current pulse) allows the discharge at the residual current pulse in the following half cycle to inherit the spatial discharge structure at the major current pulse in the previous half cycle. As a result, a complementary evolution of spatial discharge structure is observed in a whole period, showing us a nested complementary pattern.10 cm−3) in the gas gap, which dominate the spatial discharge structure due to the “spatial memory effect” and allow the residual discharge to maintain the similar spatial discharge structure as that of the previous major discharge. As we know, when the electron density in the gas gap holds at the level of no more than 109 cm−3, the electron density increases linearly from the cathode to the anode and the space charge effect is so weak that no cathode fall is formed near the cathode. The Townsend mode appears. In this case, the discharge is dominated by the electric field in the gas gap, instead of the space charge. If the glow mode initiated at the major current pulse is not sustained until the following half cycle and the Townsend mode appears before the onset of discharge in the following half cycle, there should be no chance to initiate the residual discharge for lack of sufficient space charges remained in the gas gap. Therefore, the electron density should be more than the order of 109 cm−3 after the decay in this phase from the major current pulse to the subsequent residual one. In addition, the electron density will continue decreasing after the residual discharge mainly due to diffuse when the external voltage is still low. If the electron density still holds at the level of more than the order of 109 cm−3 closely before the next major current pulse and no Townsend mode appears, the space charge effect would be great enough to dominate the spatial discharge structure throughout the whole period. The spatial discharge structure will approximately maintain the same in the positive and negative half cycles and no complementary evolution of spatial discharge structure can be found. Thus, the electron density should be reduced to the order of 109 cm−3 closely before the next major current pulse.The nested complementary evolution of spatial discharge structure each period is essentially related to the residual discharge. In the phase from the major current pulse to the subsequent residual one, the electron density in the negative glow region, where the electric field almost drops to zero and the electron density reaches its maximum, is gradually reduced mainly due to the diffuse of charged particles. However, discussion above suggests that the glow mode is always sustained between the major current pulse and the subsequent residual one. This means that there still exist a large number of space charges  is only considered and a plane source model is employed to approximately characterize the decay process of electron density in a one-dimensional coordinate. The electron density can be expressed asκ is an arbitrary constant, De is the diffusion coefficient, and ne0 is the initial electron density just after the major current pulse. The coordinate origin in a non-static coordinate system is considered at the spot where the electron density reaches its maximum in the negative glow region. The electron density at the origin is simplified aswhere t, which is closely associated with the driving frequency f. When the driving frequency is too low, the interval from the end of discharge in the previous half cycle to the onset of discharge in the following half cycle is so long that overmuch time is available for the electron density to decay to the order of 109 cm−3 or even lower. The residual discharge is not initiated due to few space charges remained. On the other hand, when the driving frequency is too high, there is no enough time for the electron density to decay to the order of 109 cm−3 before the next major current pulse arrives in the following half cycle. The discharge fails to transform from the glow mode to the Townsend mode before the next major current pulse. In the two cases, the nested complementary pattern is not formed. Thus, the driving frequency is required to be limited in a certain range to observe the nested complementary pattern.As addressed above, to obtain the nested complementary evolution of spatial discharge structure, there should be a residual discharge in glow mode and a subsequent discharge in Townsend mode before the major current pulse each half cycle. The equation t is written as 9 cm−3 under two typical conditions. When the frequency decreases to the one at which the residual discharge exactly occurs and ne falls to the order of 109 cm−3 at the decay time tlow soon after the residual discharge, it corresponds to the lower limit of driving frequency flow. When the frequency increases to the one at which the Townsend discharge appears with ne reduced to the order of 109 cm−3 at the decay time tup closely before the next major current pulse, it is considered to be the upper limit of driving frequency fup. It should be noted that the lower and upper limits of driving frequency cannot be numerically figured out, due to the complexity and incompleteness of the fluid model employed here. Thus, more simulations were carried out at other driving frequencies to roughly determine the driving frequency range. It is found that the nested complementary pattern appears at the driving frequency varied from 17 to 21 kHz. When the driving frequency declines below 16 kHz or rises above 21.5 kHz, the nested complementary pattern cannot be observed. Two extra examples are given below to address this issue.Here, the decay time flow), the waveforms of the external voltage and discharge current over one period are illustrated in 9 cm−3 or even lower before the onset of discharge in the following half cycle. The electron density radial distributions at the major and second current peaks both in the positive and negative half cycles are depicted in M1 = 599.9 μs) in the positive half cycle, while M3 = 639.0 μs) in the negative half cycle. It is found that the electron density has the maximum in the center and drops along the radial direction in both the two major current peaks, where a similar or same center-advantage distribution is observed. As seen from M2 = 607.8 μs) in the positive half cycle is quite similar or the same as that at the second current peak (M4 = 647.9 μs) in the negative half cycle. Here, a concentric ring pattern is indistinctly presented. Further simulations suggest that no Townsend mode appears between the major current pulse and the second one per half cycle. This is responsible for the failure in the formation of a complementary pattern at the two adjacent current pulses in the same half cycle. Comparison of the electron density radial distributions in the positive and negative half cycles indicates that the discharge at the driving frequency of 12.5 kHz has the similar or same spatial structure in the two adjacent-half cycles and no complementary evolution of spatial discharge structure is observed. The nested complementary pattern is not formed.With the driving frequency decreased to 12.5 kHz  in the positive half cycle (see K3 = 114.87 μs) in the negative half cycle (see K2 = 108.63 μs and K4 = 116.96 μs) in the positive and negative half cycles respectively. It is found that the discharge has the similar or same spatiotemporal behavior in the two adjacent-half cycles and the nested complementary pattern disappears at the driving frequency of 60 kHz.With the driving frequency increased to 60 kHz, the waveforms of the external voltage and discharge current over one period are shown in ycle see  is very ycle see . Figure When the driving frequency varies from 21.5 to 60 kHz, period multiplication and chaos phenomena are distinguished from the temporal evolution of the discharge currents, which are not presented here. This behavior is attributed to the mismatch of different time scales that are important in electron production and lossGenerally, it is difficult to realize a diffuse glow-like/Townsend-like DBD in argon at atmospheric pressure in experiments. Our simulation results predict the nested complementary pattern in the atmospheric-pressure argon DBD for some special conditions, under which the glow-like/Townsend like discharge is achieved. This pattern formation is explained by the charge transport regime. It is found that this unique behavior is mainly attributed to the fact that sufficient charged particles are trapped in the gas gap due to the last discharge and able to influence the subsequent discharge through the “spatial memory effect”. As for the argon DBD characterized by the nested complementary evolution of spatial discharge structure, there should be a residual discharge in glow mode and a subsequent discharge in Townsend mode before the major current pulse each half cycle. This allows the nested complementary pattern to only occur in a limited range of driving frequency.r−z) for the atmospheric-pressure argon DBD that has similar set-up to the one presented in ref. V is the external voltage amplitude fixed at 1.8 kV and f the driving frequency of external voltage with its value varied from 12.5 to 60 kHz. The z-axis is directed from the lower barrier to the upper one with the upper surface of lower barrier designated as the origin.For this study, a two-dimensional (2D) self-consistent fluid model has been developed in an axisymmetric cylindrical coordinate (+), and metastable atoms (Ar*). Reactions including excitation, direct ionization, stepwise ionization, deexcitation, radiation, and electron-ion recombination in the bulk of gas gap are taken into account. All the reaction coefficients are referred to those used in refs. In this model, the 2D simulation domain includes the two dielectric barriers and the discharge gap  meshed finer than the middle in the gas gap. To improve the efficiency of the simulation code, an adaptive time stepping is used with the initial time step of 10−13 s. It should be noted that all the simulation results discussed in this work are presented when the discharges are fully stabilized.How to cite this article: Jiang, W. et al. Prediction of nested complementary pattern in argon dielectric-barrier discharge at atmospheric pressure. Sci. Rep.5, 16391; doi: 10.1038/srep16391 (2015).


These associations between chemical and biological similarity are studied as discontinuities in activity landscapes. Particularly, activity cliffs are quantified by the drop in similar activity of chemically similar drugs. In this paper, we construct a landscape using a large drug-target network and consider the rises in similarity and variation in activity along the chemical space. Detailed analysis of structure and activity gives a rigorous quantification of distinctive pairs and the probability of their occurrence.Highly chemically similar drugs usually possess similar biological activities, but sometimes, small changes in chemistry can result in a large difference in biological effects. Chemically similar drug pairs that show extreme deviations in activity represent s) and variation (d) in activity of drugs on proteins. Interactions between drugs are quantified by considering pairwise s and d weights jointly with corresponding chemical similarity (c) weights. Similarity and variation in activity are measured as the number of common and uncommon targets of two drugs respectively. Distinctive interactions occur between drugs having high c and above (below) average d (s). Computation of predicted probability of distinctiveness employs joint probability of c, s and of c, d assuming independence of structure and activity. Predictions conform with the observations at different levels of distinctiveness. Results are validated on the data used and another drug ensemble. In the landscape, while s and d decrease as c increases, d maintains value more than s. c ∈  is the transitional region where rises in d are significantly greater than drops in s. It is fascinating that distinctive interactions filtered with high d and low s are different in nature. It is crucial that high c interactions are more probable of having above average d than s. Identification of distinctive interactions is better with high d than low s. These interactions belong to diverse classes. d is greatest between drugs and analogs prepared for treatment of same class of ailments but with different therapeutic specifications. In contrast, analogs having low s would treat ailments from distinct classes.We analyze pairwise similarity (d along the axis of c represent canyons in the activity landscape. This new representation accounts for distinctiveness through relative rises in s and d. It provides a mathematical basis for predicting the probability of occurrence of distinctiveness. It identifies the drug pairs at varying levels of distinctiveness and non-distinctiveness. The predicted probability formula is validated even if data approximately satisfy the conditions of its construction. Also, the postulated independence of structure and activity is of little significance to the overall assessment. The difference in distinctive interactions obtained by s and d highlights the importance of studying both of them, and reveals how the choice of measurement can affect the interpretation. The methods in this paper can be used to interpret whether or not drug interactions are distinctive and the probability of their occurrence. Practitioners and researchers can rely on this identification for quantitative modeling and assessment.Intermittent spikes in  The structural features of a drug compound describe its physicochemical properties that determine its biological activity on protein targets –4. A vasdistinctive interactions using these measures and try to interpret them.Medicinal chemistry has relied on the similarity principle –6. It imtopography is a two-dimensional space of chemical structure and activity to which a pair of drugs can be mapped. In any landscape, the activity cliffs are represented as discontinuities occurring due to distinct drops in similar activity of chemically similar drugs. These are distinctive drug interactions studied with great interest. Here, we analyze them using topographical features of the landscape. Researchers have studied activity cliffs using graphical methods of interpreting the landscape. They have given detailed specifications of cliffs [Activity landscape of drugs or the pharmacological f cliffs –12. The f cliffs , 12 and f cliffs . Quantitf cliffs . In prevf cliffs .s and variation, d) jointly with the corresponding chemical similarity c to investigate distinctiveness. The measures are proportional to Jaccard index and represent weighted networks of drugs [s refers specifically to the number of commonly active targets of two drugs. Using probability distributions of s, d, c we identify medicinal categories involved in the highly distinctive interactions. The predicted probabilities of s and d jointly with c are computed. This gives the probability of distinctiveness at varying levels. The level of distinctiveness of a pair of drugs (or drug interaction) is specified according to the magnitudes of the attributes s, d, c for that pair. We find that the choice of the measurement (low s or high d) affects the interpretation of distinctiveness and the topography. These measures could impact the way in which we interpret the relation between two drugs in terms of their selectivity. Drug selectivity is a property exhibited by a drug when it is active on a protein as opposed to different proteins [s and d can indicate how commonly and uncommonly selective the pairs of drugs are, but this is not necessary.This paper attempts to provide a mathematical basis of the extreme deviations observed. It would facilitate identification of distinctive interactions on different scales of measurements. The paper introduces a probabilistic analysis of the pharmacological topography and uses two measures of activity  corresponds to a divergence of both the nature and size of their activity profiles. Plausibly, a slight change in choice of molecules results in a significantly dissimilar analog of a drug while not altering its functioning on many targets. Conversely, the activity profiles of new medicines may be vastly different from those of their chemical analogs. The propensity of distinctiveness in drug interactions has implications on drug substitution. The discovery of multi-target drugs is beneficial for drug repurposing and it can result from small structural changes, which would affect distinctiveness of pairs of drugs. The method of analyzing distinctiveness is, therefore, important. Through a rigorous quantitative assessment of the activity landscape, this analysis aims to contribute not only to drug design but also to the decision making of the pharmacologists. Practitioners can apply these methods and criteria to classify the drug interactions as distinctive.Highly distinctive interactions can be filtered as structurally similar drugs having either the most variant or least similar activity profiles. Intriguingly, there is little consensus on the distinctive interactions filtered with the two measures. We study the diversity of pharmacological space using a large network of drugs and protein targets. The similarity principle is contradicted in the region where cliffs are present, however, the sizes of these cliffs are variable . Structud is much higher than that of s. Quantitative comparison of the landscape reveals that structural and biological similarities, two a priori independent manifestations of drug interactions, are unevenly associated and maybe complementary. The presence of distinctiveness is highlighted by both s and d. As chemical similarity increases, both measures decrease. However, the decrease in s is more significant than that in d, as d maintains its above average value. We show that activity variation is a more suitable measure for characterization of the pharmacological interaction space than activity similarity. Intermittent rises in d with chemical similarity maybe interpreted as activity canyons or gorges of varying levels. In this representation, the rises in both measures can be assessed relative to their respective ranges of magnitudes and compared across all pairs. The rises in activity similarity and variation are interpreted as increase in non-distinctiveness and distinctiveness respectively, characterized as activity canyons instead of cliffs. Therefore, distinctiveness is a general term that may represent abrupt deviations in the landscape as cliffs or canyons depending on the method of measurement. Moreover, the canyon representation allows for the quantitative analysis of ‘how distinctive and how probable?’, demarcating the pharmacological subspace for finding distinctiveness. These rises in activity variation are probable all along the chemical space and more distinct than drops in s. The predicted probability model introduced in this paper helps in identifying distinctive interactions and the probability of their occurrence in the landscape. It also helps in identifying distinctive drug associations that are not only significant and rare but also less apparent.In the present pharmacological space, distinctive interactions are more probable than non-distinctive interactions. Non-distinctive interactions are regular as they conform to the similarity principle. Distinguishing among the distinctive interactions requires precision. The range of s) or the number of proteins on which the activity of two drugs differs (d). Thus both s and d are proportional to the Jaccard index used for calculation of chemical or structural similarity c. “c used for this analysis.We analyzed the bipartite graph of 1354 drugs and 1596 proteins as their targets listed in the Drugbank database. In this graph, a link between a drug and protein signifies that the drug is active on that protein. An interaction between a pair of drugs indicates their association or connection based on whether or not they have a common target. For further analysis of measurements of similarity and variation in activity between drugs, we use information on activity of drugs on proteins to construct a weighted adjacency matrix. Every element of this matrix gives the pairwise weight measures of the interactions. An interaction is quantified in terms of the number of common targets  found above average for s and d across each row of plots in Fig. d is more abundantly occupied by points than the corresponding region for s. This implies that a pair of structurally similar drugs is more likely to possess highly variant than highly similar activity.We examine how the measures vary on all pairs of drugs in different ranges of chemical similarity, canyons or gorges depending on the steepness. This also applies to s but less distinctly.Intermittent spikes in activity variation occur in the intermediate range of chemical similarity f such that—change in physiological activity = f (change in structure) [s or d) is considered for quantifying biological activity. We are interested in the two dimensional interactions relating structure (c) and activity (s or d). Measures like SALI have been applied to quantify activity cliffs [c (and low d) or low c (and high d). The overall score can be misleading for identification of distinctive interactions with high c and high d. We therefore present a probabilistic analysis of the each of the measures s and d jointly with c. This informs us not only the significance of cliffs but also where  they are found in the chemical space. In the canyon representation above, intermittent rises in the measures characterize distinctive interactions. The main purpose of this representation is for finding the probability of occurrence of various levels of (extreme) deviations. In other words, for every pair of drugs, it quantifies: how distinctive and how probable?In general, medicinal chemistry has relied on quantification of structure activity to obtain a function ructure) , 4. Herey cliffs . Howeverc and s) and (c and d). This analysis is based on the distributions of c, s, d weights representing all pairs of drugs. Figure s), P(D = d), P(C = c). While s and c conform to a power law probability distribution, d follows an exponential distribution indicating that on the axis of d, the probability of high magnitudes diminishes greatly. This indicates that probability of finding extremely large variation (d) diminishes much more than the probability of finding very large s weights, however, the magnitudes of d are much higher than those of s when c, s indicate that the variance of c is much higher than that of s, implying that s is more or less homogeneously distributed in the interaction space. “We compute joint probability of structural similarity with activity variation and with activity similarity, that is, the probabilities of  and 7.44 (7.01) respectively. We compare the two measures by computing the predicted probabilities of s and d above their respective averages j = 0, 1,…, 9 for d are almost consistently and significantly higher than those for s. When they are not higher, the difference is insignificant. We also compare the empirical probability of standardized scores of s and d being positive (above their averages). The probability of finding positive values of d is higher than that of s. It must be noted that c is considered at least equal to 0.3 because c < 0.3 is not significant for any analysis. Ignoring all pairs with c < 0.3 in this data, we have We find that the observed average (standard deviation) of n= 0, 0.2, 0.4,…, 9. We use the averages and standard deviations for the entire  data and compute RMSE (see “ppred) and observed probability (pobs) for s and d. This is as low as 0.0001 and 0.0006 for calculations of d and s respectively. It is remarkable that while generalized predictions are made with the assumption of independence of s and c, and, d and c on all pairs of drugs, they confirm the observed dominance of above average d values. Figure d.We validate the predicted probability formulas  are higher in magnitude than 75% of below average s weights  and biological (s or d) properties can be considered as independently generated for studying the discontinuities in the landscape. A distinctive interaction can be measured either as high c, d or high c and low s. The almost contrasting construction of both measures results in a much higher range and distribution of magnitudes of d than of s. This affects the quantitative assessment of the interaction landscape, particularly for distinguishing structure–activity functions displayed by drugs. We establish using probability distributions of interaction attributes that it is less significantly likely for a pair of drugs with high structural similarity to act similarly than oppositely on a target. Further, the decrease in similar activity along the chemical space lacks the steepness of a cliff. In the given pharmacological space, high d dominates the chemical space in terms of not only magnitudes but also the probabilities. This behavior is mostly observed in the intermediate range of c, which we refer to as a transitional regime as it marks the increasing significance of d.The interpretation of the topography of pharmacological interaction space can vary according to the choice of measurement of c increases, interactions corresponding to high d are relatively rare and more distinctive than those of low s. This facilitates preparation of analogs for intricately specified treatments. Low s interactions can occur between drugs affecting different classes. However, distinctive interactions of high d are rarer and occur between structural analogs aimed to treat same kinds of ailments with increasingly intricate specifications. Such analogs providing advanced or versatile treatments of same kinds of ailments vary highly in their activity profiles. It is important to note that such associations are identified with high activity variation instead of low similarity.As Given the complex nature of structure–activity relationship study, a perfect conceptualization of the pharmacological topography may be hard. This analysis aims to suggest an alternative characterization resulting from mathematical and statistical formulations. A crowded valley of magnitudes with intermittent spikes of variation in the chemical subspaces, may be characterized as activity canyons or gorges. In a way this can be an encouraging sign for diversity in drug development. For instance, if a minor change in structure gives rise to an analog, then the two may be active on different kinds of targets more often than not. The sizes of their activity profiles may also be quite different. The methods provide a basis for practitioners and pharmacologists to identify distinctive interactions and the region of chemical space that they are probable to be observed in.Ai(k) = 1 if a drug i is active on protein k and 0 if the drug is not active. This constitutes the activity profile of drug i. Chemical similarity of drugs is computed pairwise with tanimoto similarity method [N = 1354 drugs, C, has Cij give the chemical similarity of the pair of drugs and We use categorical binary activity results of 1354 drugs on 1596 protein targets. The activity information is given as binary attributes, y method . The weiS—If Ai and Aj are activity profiles of drugs i,j, that is the proteins k for which Ai(k) =1 and Aj(k) =1, then S has s. The commonly unresponsive proteins are ignored. Every element of the matrix denotes the weight of the connection or the interaction between pairs of drugs.D—Variation between the activity profiles of two drugs measures the number of differences in their binary attributes, that is the number of proteins on which they differ in activity. The weighted network D of pairwise distances of drugs S as we ignore the commonly unresponsive proteins on all pairs of drugs while computing S. Both these measures [S and D so as to make their magnitudes and variances distinct, hence we compare the actual numbers of similar and variant activities.measures , 16 are s), variation (d), chemical similarity (c), we write the mathematical forms of the normalized probability distributions asp shown on the vertical axis. The first two top figures for c and s are on a log–log scale, so that the straight lines are given by smin, cmin, dmin correspond to the minimum values of biological, structural similarity and variation respectively. We have dmin = 0 and for a well defined power-law distribution, we consider smin = 1, cmin = 0.004.Assuming continuous distributions for the pairwise weights as random variables for similarity  and P(s) from above, the joint probabilities approximate toP(c) and P(d), we obtainSimilarly, using probability distributions of Solving Eqs.  and 2)  2) we obEquations  and 4)  4) give s as d as smin = 1, cmin = 0.004, dmin = 0, We compare the two measures by computing the predicted probabilities of similarity and distance above their respective averages. We denote the average and standard deviation (respectively) of ing Eqs.  and 4)  s as \dos with d above their respective averages, we measure the predicted probability of finding n=0,1, 2… respectively, in the same chemical space. The standard deviations may also be increased in steps of 0.2. These are considered as measurements for the categories s and d are subtracted from their averages and divided by their standard deviations for a certain chemical space. Then the comparison is between the probability of finding s and d above their average = 0 in a given chemical space.In order to compare c and s (or d). Thus c then P(c) = 1.s and d, we apply two tests.In order to test the significance of the difference between the observed probabilities associated with s and d above their averages in the same chemical region We use this test to compare the relative frequencies (or probabilities) of s of the  two measIf 0: H1: H0 in favor of H1 at the level of significance We reject Hn = 0, 1, … 8. We also sample the observations by increasing standard deviations in intervals of 0.2. All the probabilities so computed can be generalized as s, d. In this way, we consider the variation of probabilities in the space above the average sampled at increasing number of standard deviations.This test is used to compare the average of the difference between probabilities 0 of equality of probabilities on average against one-sided alternative H1We test the null hypothesis H0: H1: 1: Hn − 1 d.f. Here p1 − p2, n = 9 here. The null hypothesis of no difference is rejected in favor of alternate H1 if the observed t is greater than the critical value of test statistic i.e. t > tc . We apply this for comparing probabilities of s and d in different chemical regions.with the test statistic,


Contrast this to the Vogel-Fulcher law log η  = B + A/(T − Tc). A notable feature of our description is that even though the location of the equilibrium second-order transition in temperature-pressure space is given by the break in the entropy or volume curves the viscosity and its derivative are continuous through the transition. The new expression for η  has no singularity at a critical temperature Tc as in the Vogel-Fulcher law and the behavior reduces to the Arrhenius form in the glass region. Our formula for η  is discussed in the context of the concepts of strong and fragile glasses, and the experimentally observed connection of specific heat to relaxation response in a homologous series of polydimethylsiloxane is explained. The frequency and temperature dependencies of the complex viscosity η (ω< T), the diffusion coefficient D(ω< T), and the dielectric response ε (ω< T) are also obtained for our kinetic model and found to be consistent with stretched exponential behavior.A kinetic theory of glasses is developed using equilibrium theory as a foundation. After establishing basic criteria for glass formation and the capability of the equilibrium entropy theory to describe the equilibrium aspects of glass formation, a minimal model for the glass kinetics is proposed. Our kinetic model is based on a trapping description of particle motion in which escapes from deep wells provide the rate-determining steps for motion. The formula derived for the zero frequency viscosity  Sc, or at least the entropy reaching a critically small value, is associated with glass formation. Sec. 1.4 describes briefly the many experiments that support the entropy theory of glass formation. Sec. 1.5 offers a critique of equilibrium theories. In Sec. 1.6 the suggestion is made that the Sc = 0 criterion can be replaced by Sc = Sco. Sco is a small critical value of the entropy which is dependent on the time scale of the experiment but is positive even for infinitely long time scale. Sec. 1.7 contains qualitative insights into the kinetics of glass formation arising from the Sc → 0 criterion, while Sec. 1.8 makes the observation that the fluctuation-dissipation theorem provides quantitative insights into the connection between the equilibrium and kinetic properties of glasses.In this paper we first critically review the entropy theory of glasses. After defining a glass in Sec. 1.1 we show in Sec. 1.2 the need for an equilibrium thermodynamic theory of those materials that form glasses. Sec. 1.3 gives our reasons for believing that the vanishing of the configurational entropy D and complex viscosity η , while in Sec. 2.5 frequency dependent D(ω< T), η (ω< T) and the dielectric response ε (ω< T) are obtained. These quantities each depend on the distribution of well depths W(E). This quantity W(E) = exp (Sc/k) is discussed in Sec. 2.6 where viscosity is shown to be a function of free energy. Finally, Sec. 3.1 discusses our results while Section 3.2 offers some conclusions.The kinetic theory is developed in Sec. 2. In Sec. 2.1 we pass from phase space to configuration space and gain an insight into the topology of configuration space. In Sec. 2.2 we use the principle of detailed balance to evaluate the transition rate constants of the master equation describing minimal models of glass formation. In Sec. 2.3 using a trapping model for the phase point we define these minimal models and derive their associated (master) equations. Sec. 2.4 contains derivations of the zero frequency diffusion coefficient V, and entropy S, fall out of equilibrium as we lower the temperature past some temperature Tg which depends on the rate of cooling. Above Tg the relaxation times associated with viscosity are less than the time scale of the experiment, while below Tg they are greater. The above definition describes the formation of a crystal as well as a glass so we augment our definition by requiring that the extensive thermodynamic quantities be continuous at Tg and that there be no change of spatial symmetry as we cross Tg. This operational definition immediately suggests a number of questions which must be answered if we are to understand glasses. 1) What are the V and S equations of state on the high temperature side of Tg? 2) For a given rate of cooling, why does the glass transition occur at one temperature, Tg, rather than some other temperature? 3) What are the thermodynamic properties well below Tg where the relaxation times for diffusion of molecules are so long that some degrees of freedom are frozen out and only oscillatory motions occur? Experimentally the glass is known to behave like an elastic solid. 4) What is the viscosity η , where ω is frequency? The first three questions are concerned exclusively with the equilibrium properties of glasses.We define a glass to be a material which is an ordinary liquid at high temperatures and whose thermodynamic extensive quantities, volume Tg and well below Tg. It is sensible to ask what they are.Glasses have equilibrium properties above The crystal phase is not ubiquitous. This proposition was proved in Ref. . TherefoT2 to which the Tg tends in very long time-scale experiments.An equilibrium theory is needed –5 to resAn equilibrium theory is a necessary prerequisite for an understanding of the kinetics .There are four bona fide reasons to formulate an equilibrium theory of glasses , 2. TheyV and S. It is required, of course, that the important characteristics of the molecules be taken into account, at least within a minimal model. This minimal model (the simplest model which retains the essence of the problem) must have both intermolecular energy to allow for volume changes and intramolecular energy to allow for temperature dependent shape changes of the molecules. The lattice model of Gibbs and Di Marzio . In 2) the glass temperature is predicted to rise as we lower molecular weight of rings in accordance with experiment. This is purely an entropy effect  where z is the coordination number of the lattice. In the Gibbs-Di Marzio model this distribution is realized above T2, but at lower temperatures each chain is frustrated by its neighbors from achieving its Boltzmann distribution of shapes. Instead, the distribution that existed at T2, P2 persists as we lower the temperature at constant P2. The number of holes also remains constant below T2 while in the Milchev theory it continues to decrease. The fact that experimentally the volume versus temperature curve for a glass parallels the volume versus temperature curve for a crystal supports the view that the number of holes is constant below Tg. It must be mentioned however that some computer calculations exist that support the Milchev formula . The process of diffusion as well as the process of flow can be viewed as a jumping out of a deep well, a subsequent wandering about the phase space between deep wells and a dropping into a deep well different from that from which it had exited. The process then repeats itself. Obviously this process becomes more infrequent at lower temperatures resulting in increased viscosity and decreased diffusion. There are several reasons for this. First, the deeper the well the longer the time to escape it—wells are effectively deeper as we lower temperature; second, the further apart the wells are the more time it takes for a phase point to wander from one well to another; third, the further apart the wells the larger the probability that the wandering phase point will fall back into the well it has just escaped resulting in no net flow. See Thus, as we lower the temperature the number of configurations D to the autocorrelation function of the particle velocity v. More generally the F–D theorem relates χ  arising from an impulsive force at ) to the correlation in fluctuation at these two space-time points [η  will show a discontinuity as a function of T, P since the volume and entropy do. Similarly, for a system undergoing a second-order transition we can expect that the viscosity will show discontinuities in slope since the volume and entropy do (see later in this paper however). There are many examples in the literature of dissipative quantities such as viscosity, diffusion coefficient, electrical conductivity, particle conductivity and thermal conductivity which show breaks as a function of temperature as we pass through the glass transition. However, it is also true that a genuine falling out of equilibrium will also cause the same kind of behavior. It is uncertain how one distinguishes between the two effects. Movement of the transition point as a function of the time-scale of the experiment seems not to be a distinguishing characteristic since this happens also for systems known to have genuine first-order transitions-supercooling being an obvious example.Whenever there is a thermodynamic phase transition the fluctuation-dissipation (F–D) theorem  suggestse points . Since tMore generally the frequncy dependent diffusion coefficient is given byE′(.. jq..) appears in the partition function Qqj, jp are the generalized position and momentum coordinates of the N particles, K is the kinetic energy, E′ is the potential energy and Λ the thermal wavelength. Since the kinetic energy is quadratic in ip the integration over ip is straightfoward. In polymers, even if E′ is pairwise additive, E is not [K are in general dependent on iq. The simplification of Tg from above the wells effectively become very deep because of the 1/kT term. One then talks about flow as a motion from one deep well to another deep well via the higher energy continuum.The potential energy surface of a liquid E is not  because N identical noninteracting hard point particles on a line of length L. The partition function is given byHere, however, we wish to emphasize a different aspect of the phase space topology. Consider the configuration space of NL. Now consider the case where the particles each have a diameter d. The partition function isϕ, being the volume fraction occupied by the particles, is on the order of 1. Since N is on the order of Avogadro’s number we see that the fraction of the volume of phase space occupied by the extended particles is infinitesimally small relative to the unconstrained particles. Based on this picture a point in configuration space wanders on the finest of gossamer threads [N-dimensional hypercube of phase space as a fine network whose total volume is an infinitesimal fraction of NL. The application to glasses is in the observation that as we lower T the effective value of d increases, resulting in even fewer and finer gossamer threads for the phase point to travel on. Thus, not only are the number of paths (threads) between two phase points fewer as we decrease T but also as one traverses a given thread the potential energy minima are effectively deeper and the barriers effectively higher.The volume of configuration space for this system is given by  threads  which peThe above discussion serves to show how important it is to know how the deep wells are connected to other wells. In order to solve this problem we need to construct a model of the topology of configuration space and to calculate the transition rates for jumping from well to well within this model.iE/kT) as the fraction of time that a system spends in state i but it does not say how often the system jumps from state i to state j. To determine this we use the principle of detailed balance in the formijα is the rate of jumping from state i to j and iN is the fraction of time a system spends in state i. In using τ where τ is the average time to exit the well. If we also recognize that the probability of jumping out of the well is exponential in time [P is the probability density of exiting the well at time t. It is imagined that once the phase point has escaped the well it wanders around in the configurational sea of the high energy region of phase space until it falls into a low lying well, starting the flow process all over again. This configurational sea consists of many shallow energy wells, so it is expected that jumping out of the deep wells are the rate determining steps.Boltzmann’s law gives exp is proportional to the well depth. These vertical lines represent many possible paths in configuration space leading to the deep well. In It can also be shown using the methods developed previously  that FigHowever, The results of this paper will allow us to conclude that although α) or left (β) and the Roman symbol rate constants denote stepping down (b) or up (A).The Master Equations describing the minimal model of jf is the fraction of systems in state j and jαr is the rate of jumping from state j to r. Multiplying the first equation by f′k and the second by jf we obtainℓ ≡  and i ≡  then we obtainiN, of the composite system is the product of the complexions, jf, of the individual systems, but the composite transition coefficients are sums of the individual transition coefficients. These results are readily generalized by the process of induction to a system consisting of any number of subsystems, the only condition being that the subsystems do not interact with each other. We again see that the complexion of the composite system is the product of the complexions of the individual systems, but the composite transition coefficients are sums of the individual transition coefficients.We have already shown in Sec. 2.1 that one can integrate over all the momentum variables of phase space so that we deal only with position variables (configuration space). We would like to go further and deal with the smallest number of position variables possible. We begin by supposing that there are two separate noninteracting regions of space each with their own master equationsThus, if we could find a smallest set of independently interacting molecules we will have simplified our problem considerably. Fortunately there is a confluence of intuition and experiment that suggest that this can be done. First, what is happening at point a cannot be influenced by what is happening at point b provided that the two points are sufficiently far apart. So, there is a smallest size. Second, this size seems to be very small indeed. Stillinger, on the basis of computer modeling and other considerations has concluded  that thei, MiN of As a result of these considerations we can maintain that the D, v, b, and A can all be position dependent.Ej is the depth of the well. We argue that the energy appears only as a barrier restricting the escape from the wells-there is no attraction of the phase point into a well. The jb are also all chosen to be equal because we can think of nothing that distinguishes them from each other. Allowing the jα to be different from the jβ accounts for a drifting of the phase point towards a region of phase space. This should be useful if we impose an external field. If we assume no x dependence for α and β then D and v are constants and the ∂2DN/∂x2 term is the ordinary diffusion term. Our Equations now readt, x dependencies explicitly.The rate constants are determined as follows. From α and β are much greater than b we know that after jumping out of a low lying well the phase point will travel extensively horizontally before being captured by a deep well. Since b does not depend on x and is not a function of well depth the rate of filling the wells is random. Thus the horizontal distribution of well depths which we assume to be random along the chain (see W(E) is the number of wells of depth E then they are filled with a rate proportional to W(E). Over a large period of time the escaping from wells is determined by both W(E) and the rate of escape (exp(− β |E |)) from individual wells. This allows us to replace the distribution of wells by wells of one depth. In this case A now has no x dependence. Using the method of moments on Since hain see  is unimpnth pair of equations for the nth order moments in terms of the lower order sets. We will exploit this fact in the next section.The nice thing about these equations is that we can solve the jb = b, ΣjN = nN, and assuming that horizontal motion is so fast that jN = N, the sum over the jN in Finally, considering only the sequence in time of the occupation of the deep wells by the configuration point, with N being the amount of material in the mobile phase, while M is the amount of material adsorbed on the adjacent surface or in pores [One notes that in pores .η *(ω< T), diffusion coefficient D(ω< T) and dielectric response ε (ω< T).Our minimal models are all now well defined and deriving their implications is merely a matter of mathematics, albeit sometimes very difficult mathematics. The remaining conceptual problem, to which we now turn, is to relate the solution of these minimal models to the frequency and temperture dependent complex viscosity A/(b + A) (because from A/b = exp(− |jE |/kT) and the wells are deep we will ignore the A in the denominator of A/(b + A)). These equations have the obvious interpretation that everything, both drift and diffusion, is being slowed down by the factor A/b which is the ratio of jump rates. As long as the particle is in a deep well there is no activity. Any resulting activity is proportional to the rate of escape, exp(− |jE|/kT), from the deep wells.Notice that the diffusion coefficient is diminished by the factor ∂N/∂t = D∂2N/∂x2) has as its Green’s function the Gaussian distribution (4πDt)1/2exp(− x2/4Dt). In the probabilistic formulation of the diffusion equation this Green’s function has the physical interpretation of representing a random walk as in D while the pausing corresponds to the time spent in the deep wells. Thus, the effective diffusion coefficient isWe now seek to further interpret this result. The ordinary diffusion equation without sinks . Since viscosity is inverse to diffusion we will assume that the viscosity is proportional to the average time spent in the deep wells. This notion is verified in P is the normalized probability density that the configuration point escapes the well at time t.In P is easily calculated. The probability ψ of the particle being in the well at time t isP for exiting the well at time t isWhen all the wells are of the same depth ψ is a good one. To see why consider the configuration point in a well. It decays exponentially initially. This can be seen by solving the generic matrix equation to which the set 27 belongsN is the set  and A is the matrix of The exponential approximation for P for this latter case and also calculate a new effective diffusion coefficient. P is exactly calculable from x involved in E sees only the barrier and therefore the probability that it be in the well at time t is given by W(E) be the weight distribution for wells of depth E. Notice from ∫W(E)Pdt = b−1W(E)exp(|jE/kT) which states that the time spent in wells of level E is given by the Boltzmann factor weighted by the degeneracy factor W(E). This is in perfect accord with the ergodic theorem. An estimate of the relaxation function p describing the exiting from wells can now be made by weighting the distribution function P (see E by the weighting function W(E).However, solving the problem where the deep wells are all the same depth is not the same as solving the problem for glasses since glasses have a distribution of well depths. We need to evaluate t,T) see  for the The right-hand-side of M particles that have simultaneously jumped out of their wells in order to have flow. It is shown in In Secs. 2.4.2 and 2.4.3 we presumed that the process of flow could occur if only one particle jumped out of its well. But suppose it is required that within a space of a given volume there needs to be η asB and M are considered to be constants.This allows us to express the temperature dependence of Aeff given byWe now seek to calculate the diffusion coefficient when we have a distribution in well depths. The answer to this can be obtained by solving iW appear as shown because the number of times a particle falls into a well of depth iE is given by iW. The argument for this is that as soon as a configuration point escapes its well, because of the large value of D while running about in the upper wells it has exposed itself to the other wells, and because b is independent of x it falls into each well with equal probability. If the number of wells of depth j is jW the configuration point falls into a well of energy jE with a probability jW and then tries to escape with a probability proportional to jA. Thus, we know that jW is proportional to the number of well of type j and the effective diffusion coefficient for The form of g with the integrand of n would be permitted if the sole criterion were that the integral equal τ. Formulated in this way it is obvious that n = 0 gives the correct g since it corresponds to a Maxwell element. Thus g is proportional to g is G0 we haveW′(E) of well depths we obtainAt zero frequency we showed thatThese relationships show clearly that non-Debye frequency behavior occurs because there is a distribution of relaxation times.ε (ω< T) can also be obtained. Debye showed that if the dipoles are each imagined to be imbedded in the center of spheres (one dipole per sphere) that are in turn imbedded in a viscous fluid of viscosity η then the dielectric response is easily calculated [η *(ω< T) then the formula is a simple generalization of the Debye formula, the only change being that η * replaces η . Thus,A is a dimensional constant. The plus sign occurs in Granted the calculation of the complex viscosity, the dielectric constant lculated . Based olculated  that if D is a constant. In order for it to have a frequency dependence we would have to have had ∫D(t−τ)∂2N/∂x2dτ for the first term on the right hand side of D(t−τ) = Dδ(t−τ). Further, from For a distribution of well depths we have as beforeW(E). Notice that if this can be done then our kinetics of glasses will depend only on the equilibrium statistical mechanics. For glasses statistical mechanics plus the principle of detailed balance is everything provided we are looking only at the long time behavior.The above relationships are quite remarkable for they state that long time relaxations-viscosity, diffusion and dielectric response depend only on the well depths and the distribution of well depths. The only thing remaining is for us to evaluate Fc = −kTlnQ which connects the configurational part of the Helmholtz free energy Fc to the partition function Q we have immediatelyThe classical and quantum mechanical partition functions are given by sse form , the Bensse form , the Avrsse form , the Adasse form  or the msse form . We discW(E) separately before we can evaluate η (ω< T). If in βF(β)) is the Laplace transform of W(E) and W(E) is the inverse transform of exp(−βF(β)).The frequency dependent viscosity, given by W(E) is given by [Another approach is to use the results of Stillinger who suggests that given by , 52W(E)W(E) is implied [The time dependent behavior of  implied .and its temperature derivative are continuous as we proceed through the transition. We had, in Sec. 1.8, used the argument that the dissipative quantities should have the same transition behavior as the thermodynamic variables. So, for a first-order transition the viscosity is discontinuous through the transition because the entropy and volume are. But we have now obtained the result that for a second-order transition the viscosity does not show a break as we traverse the transition point. In the past various groups have argued that the volume [only determinate of the kinetic aspects of glass formation in the limit of zero frequency is the thermodynamic free energy! See e volume  is the ce volume , or the e volume . We are The Vogel-Fulcher-Tammann-Hesse form  from whiThe Bendler-Shlesinger form  islogη=The Avramov form  islogη=bbs form  islogη=Tg.These forms should be compared to our form which isargument  that theB to locate the curve vertically, another of the parameters stretches the curve so that there is a fit at both high and low temperatures. Finally the third parameter gives the curve the proper amount of curvature. Viewed in this way we see that the fact that formulas of different construction give decent fits to the data is not surprising. A real test of the theories is whether they can determine the values of the three parameters from theory.Although each of the first four forms has some theoretical underpinning it is probably true that the reason they fit experimental data well is that they (the first three) are three-parameter fits and the viscosity curves are rather structureless to begin with. To see that it is not terribly significant to fit a curve of relatively little structure with three parameters imagine Fc. A real theory should contain no parameters. Schroedinger’s equation plus the laws of statistical mechanics should be sufficient. The authors intend to examine the meaning of the B and A parameters of A and B to be temperature independent.Viewed from this perspective the last two Cp,c, varies inversely with temperature [Cp,c = α/T.C is (part of) the energy of activation.Angell’s classification  of glassperature . We therCp,c = T∂Sc/∂T, Sc = ∂Fc/∂T and ignored any pressure dependence. Below The transition temperature T2 the configurational entropy is zero according to the simple version of the GD theory so that we have only energy of activation while above T2 the specific heat is assumed to decrease inversely with temperature in accord with experiment.To obtain these equations we integrated B by choosing a reference temperature T* for which the viscosity equals 1013 poise. A little algebra results inθ = CA/kT*, ζ = αA/kT*, x = T*/T. T* is the temperature for which η = 1013 poise. If we had picked 10y as the reference viscosity then the above equations would be the same with y replacing 13 and T* being the temperature at which the viscosity is 10y poise.Using T*/T decreases the curvature is larger. Below the glass temperature we predict pure Ahrennius behavior. These features are also features of Angell’s classification of glasses into strong and fragile varieties. An interesting prediction is that if T*/T2 = 1 then the initial slope at T*/T = 1 is independent of specific heat. It does however depend on C.Tg/T where Tg was defined as the temperature for which the relaxation time was one second. These curves which are reproduced in T*/T = 1 are independent of specific heat—we predict this, 2) The curvature is larger the smaller the value of Tg/T—we predict this, and 3) the curves flare out for low Tg/T with the higher specific heat (low molecular weight) material flaring up and the low specific heat (high molecular weight) material flaring down—we predict this. The filled circles are our numerical predictions. We chose A and B to fit the center curve. We then scaled ζ by the ratio of the specific heats for the low and high molecular weight polymers to obtain the upper and lower points at each temperature. Our fits assume that C is independent of molecular weight.We can test these predictions for polymers using data for polydimethylsiloxane of varying molecular weight. Roland and Ngai  using diT2. Our reason for doing this is that although the GD lattice model predicts that the configurational specific heat approaches zero as the temperature increases it does not do so with purely inverse temperature dependence. So, a combination of the two specific heat variations may better fit the experimental data.We also give the formulas for the case that the configurational specific heat is constant above T* as the temperature for which the viscosity equals 1013 poise we obtainζ = α′A/k and θ = CA/kT*. These curves again show the features of the strong-fragility plots discussed by Angell.If we again define T*/T = 1. Also the curvature of the logη vs T*/T curve becomes smaller as T*/T decreases which is contrary to the sense of virtually all experimental results.It should be noted that if either of the above forms for the entropy is substituted into the Adam-Gibbs form  one obtaT* ever be less than T2? Under the paradigm of the Vogel-Fulcher equation this is a foolish question. However, since the viscosity and its derivative are, according to T2 can not be located accurately by measurements of viscosity; we see no reason why it can not be greater than T*. The possibility that T2 corresponds to a finite viscosity may well be masked by the process of falling out of equilibrium which can be discussed only by examining the time or frequency dependent viscosity.Can C/T* varies with material should be made.The new formulas for viscosity suggests several new directions. First, an examination of the way which C the initial slope of the curve at T*/T = 1 would be inverse to T*. Also systems for which the motion is highly cooperative would show a higher C. Systems which have the same scaled potential energy surface, i.e., hE(.. iq..) where h is any constant, should display superposed fragility plots. Such systems which have no specific heat break at Tg should all superpose with the form of a straight line. Finally how the specific heat α relates to C/T* should be examined.We remark that these questions are equilibrium thermodynamic and statistical mechanical questions so that their investigation should not be difficult. For systems with constant B and A have a temperature dependence which must be added to that of the free energy. This thought is consistent with the view expressed by some that the temperature dependence of viscosity and diffusion at higher temperatures is adequately addressed by mode coupling theory and that the behavior over the full temperature range can be obtained by a cross-over treatment that combines the high temperature mode coupling theory with a theory of low temperatures such as has been presented here.Another possibility that deserves serious consideration is that the parameters We leave such a development to the future.η * and shear relaxation modulus g, the diffusion coefficient D = D and the dielectric response ε .This paragraph describes the logic of our development. We first observed that there must exist at low temperatures an equilibrium glass phase because the crystal phase is not ubiquitous. It is only for systems that can crystallize that the glass phase can be considered to be a metastable phase. We next showed that the Gibbs-Di Marzio (GD) theory –5 which Our relaxation modulus has the form of the after-effect function tabulated by Jahnke and Emde  which weη to the configurational part of the thermodynamic Helmholtz [FcMore surprising is our formula relating the zero frequency viscosity elmholtz  free eneη (T) vs T. Below the transition the behavior is Arrhenius corresponding to the fact that there is energy but not entropy of activation below the transition. Above the glass transition entropy of activation kicks in.It is surprising that the viscosity is continuous through the transition. If this conclusion holds, viscosity or other dynamical measurements may be the worst way to locate glass temperatures. The use of thermodynamic quantities which show breaks in slope should be preferred. Initially we had expected (naively in retrospect) that there should be a break in slope of Fc are used this formula displays the main features of the strong-fragile glass classification scheme proposed by Angell [η versus inverse temperature plots while glasses with large specific heat breaks show positive curvature. See Sec. 3.1.When sensible approximations for y Angell . GlassesB and A in the above equation. An approach to this problem is to excise from phase space those phase points corresponding to deep potential energy minima and solve the kinetics of such a circumscribed space. Since the resulting equations should be applicable to the high temperature side of the glass transition it may be that mode-coupling theory can be used for this part of the problem. A theory of glasses that is valid over a wide range of temperatures undoubtedly requires incorporation of vibrational properties.We have not yet examined the temperature dependence of


Unambiguous and selective standoff (non-contact) infield detection of nitro-containing explosives and taggants is an important goal but difficult to achieve with standard analytical techniques. Oxidative fluorescence quenching is emerging as a high sensitivity method for detecting such materials but is prone to false positives—everyday items such as perfumes elicit similar responses. Here we report thin films of light-emitting dendrimers that detect vapours of explosives and taggants selectively—fluorescence quenching is not observed for a range of common interferents. Using a combination of neutron reflectometry, quartz crystal microbalance and photophysical measurements we show that the origin of the selectivity is primarily electronic and not the diffusion kinetics of the analyte or its distribution in the film. The results are a major advance in the development of sensing materials for the standoff detection of nitro-based explosive vapours, and deliver significant insights into the physical processes that govern the sensing efficacy. Selective standoff detection of explosives is challenging due to the presence of volatile interferents. Here, the authors report dendrimer thin films that display distinct fluorescence responses when exposed to explosives as opposed to common interferents, allowing selective detection of nitrated explosives. The capability to reliably detect explosive vapours is essential for the provision of security, the remediation of landmine fields in current and former warzones, as well as environmental monitoring. There are a number of detection instruments available based on standard analytical techniques  but due to their inherently bulky nature and the need for swabbing to collect the sample, the use of such devices has largely been restricted to controlled screening environments and laboratories. As a consequence there is a large and unmet need for a compact portable device for infield use that enables rapid, standoff (non-contact sample collection), sensitive and selective detection of explosive vapours1345781113At a superficial level, the design criteria for sensing materials for the detection of nitro-containing explosives are ‘well established'45However, at a practical level these ‘established' criteria have a tendency to break down. For example, using energetics as the simple design criteria for a successful sensing material is not necessarily sufficient to achieve selectivity. It has been shown that the fluorescence of polymer and dendrimer sensing materials can be quenched by non-explosive materials even if the energetics for electron transfer appear to be unfavourable, leading to a critical lack of selectivity . In addiThus, although oxidative fluorescence quenching has great promise for explosive sensing, the lack of selectivity has meant that detectors based on these materials cannot be widely deployed due to the problem of interferents creating false positive responses5853  and DMNB) the luminescence is quenched while in the presence of interferents the dendrimers exhibit either no change or an increase in their luminescence . We beli3  as the e3 has a PL quantum yield (PLQY) of 67±7% in solution and 47±5% in the solid-state indicating that the first generation dendrons are sufficient to reduce the intermolecular interactions of the chromophores that would otherwise lead to the quenching of the luminescence in a film. For the study we used 4-nitrotoluene (pNT) as an analogue of a nitroaromatic explosive, DMNB as an example of an explosive taggant and naphthalene as an interferent. Naphthalene was chosen as the interferent as it is the active component in widely used mothballs, is a solid so is simple to handle yet has a high vapour pressure, is an aromatic molecule with no nitro-groups, and is readily available in the perdeuterated form required for the neutron reflectometry (NR) experiments.Dendrimer 3 films with analyte-saturated films. Despite the extensive body of literature on the detection of nitroaromatics by fluorescence quenching there have been few systematic reports on the actual solid-state sensing mechanism3 from TAS and time-correlated single photon counting, which were found to be the same within experimental error . The decay of the singlet excited state measured with TAS in films of dendrimer 3 was fitted with a sum of two exponentials with an average lifetime of 390 ps , comparing the excited state kinetics of dendrimer f 390 ps , which i3 feature a bleach signal at wavelengths shorter than ∼480 nm and a broad feature at longer wavelengths that peaks at ∼590 nm  versus ln(t), n can be determined from the gradient , which corresponds to Super Case II diffusion23where gradient . There aL is the distance travelled by the front at time t and kL is a constant. Differentiating where L/dt=0.08t0.8 nm s−1  although the thickness of the film did not change significantly.Finally, to further probe the diffusion of the analytes into the dendrimer thin films we performed NR, which allows the vertical structure of thin films relative to the substrate, that is, thickness and mass density, to be determined. One of the key advantages of NR over complementary techniques, such as X-ray reflectometry and ellipsometry, is that contrast between organic compounds with similar mass density can be achieved with selective deuterationted film  has incrr d-DMNB —the analribution . The PL ribution , as was The family of light-emitting dendrimers that shows selectivity  for nitr3 was estimated from its solution oxidation potential in combination with the PL and absorption spectra following the method by Yu et al.3, pNT, DMNB and naphthalene were −2.7 V, −1.7 V . The short centre-to-centre dendrimer-analyte distance observed by QCM is consistent with the TAS measurements, which show that electron transfer occurs on a timescale that is much shorter than that required for typical exciton diffusion lengths. It has been previously thought that relatively long exciton diffusion lengths were required to achieve effective quenchingThe QCM measurements showed that on introduction of the nitrated analytes there was a rapid uptake of mass by the sensing films. Increasing the film thickness resulted in an increase in the mass uptake, which indicates that the analytes were being absorbed by the film rather than adsorbed onto the surface. For both pNT and naphthalene Super Case II diffusion was observed. Super Case II diffusion is where a film swells as another material diffuses into it , with the diffusing material propagating as an accelerating front through the film. As such the dynamics of the analyte front are governed by the swelling kinetics of the filmFinally, the steady-state NR measurements showed that all three analytes were evenly distributed throughout the sensing film, which is consistent with the Super Case II diffusion observed from the QCM measurements. This is in contrast to a concentration gradient as was previously observed for pNT saturated bifluorene-cored dendrimer sensing filmsin situ PL measurements showed that all the analytes tested diffused into the dendrimer film via a Super Case II mechanism, whereby a high analyte concentration front propagates through the film. Critically the analyte concentration behind the front was sufficiently high such that the average dendrimer-analyte separation was much less than typical exciton diffusion lengths. Coupled with the fact that a charge-transfer state formed on the femtosecond timescale, this indicates that exciton diffusion does not play a significant role in the efficient fluorescence quenching observed. This is a key insight as it provides a new direction for designing sensing materials for the detection of vapours in the solid state, and highlights the importance of understanding the nature of the interaction at the solid–vapour interface. Finally, dendrimers containing triphenylamine centres are the first class of materials in which the response to nitro-containing analytes is dominated by the energy design criteria. The electron affinity of the naphthalene was insufficient to oxidize the photoexcited dendrimers whereas the nitro-containing analytes were able to oxidatively quench the fluorescence. Importantly, detection of the nitro-containing analytes was found not be masked by the interferent used in this work, even when the vapour pressure of the interferent was 100 times greater than that of the nitro-containing analyte , nitroaliphatic (DMNB) and aromatic  vapours could diffuse into the dendrimer film showing that dendrimer-analyte affinity was not the origin of the lack of fluorescence quenching by naphthalene. The QCM measurements combined with  analyte .et al.The solution PLQYs of the dendrimers were measured relative to a reference of quinine sulfate solution in 0.5 M sulfuric acidThe dendrimer was spin-coated onto fused silica substrates, which were loaded into a sealed optical chamber in a nitrogen-filled glove box to minimize photo-oxidation of the samples during the measurement. The measurements on the vapour-exposed films were performed with a second optical chamber with a small cap containing the analyte placed inside the chamber. Once the sample was loaded into the optical chamber the interior was flushed with nitrogen and then left to rest so that the atmosphere inside the chamber was saturated for the measurements. The broadband transient absorption measurements were performed with a HELIOS transient absorption spectrometer . The output from a Spectra-Physics Spitfire amplifier operating at 1 kHz was split to seed a TOPAS optical parametric amplifier and generate the 380 nm pump beam, with the remainder used to generate the white light continuum for the probe. The intensity of the pump beam was attenuated with a neutral density filter until the decay kinetics of the as-cast film showed no power dependence. The transient absorption data were corrected for chirp before analysis.M) from the frequency changes (Δf) of the crystal output given byThe QCM measurements were undertaken with a custom-built experimental setupf0 is the resonant frequency of the fundamental mode of the crystal, A is the area of the gold disk coated onto the crystal (0.205 cm2), ρc is the density of the crystal and μ is the shear modulus of quartz. The density of the crystal was 2.648 g cm−3 and the shear modulus was 2.947 × 1011 g cm−1 s−2 . Therefore, for an 8 MHz crystal, a 0.1 Hz change in frequency corresponds to a mass change of 0.14 ng. However, it is important to note that QCMs are very sensitive to the environment including local vibrations and temperature changes. We therefore measured the QCM drift under nitrogen at a flow rate of 1,500 ml min−1 and found that the frequency varied by up to ±5 Hz during 6,000 s running time, which corresponds to an uncertainty of ±7 ng in the measurements. It is important to note that this uncertainty in the mass is greater than the differences observed in separate measurements under the same conditions, and hence sets the upper limits on the errors. Each sorption experiment was repeated three times and average values taken for further calculation. An Xdip-SV1 dip-coater was employed for coating the QCM chip. The QCM chip was first cleaned using toluene, followed by rinsing with ethanol and acetone before being dried with a stream of nitrogen. It was then immersed in a 10 or 15 mg ml−1 solution of the dendrimer in toluene and withdrawn from the bath at a prescribed withdrawal speed to control the final film thickness. The process was done under ambient conditions. The film was then left to dry in air. Thicknesses of the films coated on the QCM crystal were determined from the optical density and absorption coefficient, and crosschecked with the thickness expected from the mass and film density−1 for the pNT, naphthalene and DMNB, respectively.where λ<18.0 Å) at the OPAL 20 MW research reactor , Sydney, Australia)λ/λ=4%) and recorded on a two-dimensional helium-3 neutron detector (Denex GmbH). Reflected beam spectra were collected at 0.7° for 1,800 s and 2.5° for 7,200–8,000 s and direct beam measurements were collected under the same collimation conditions for both reflection angles. Experiments were conducted in a custom-built sample chamber in air, which incorporated a 365 nm LED (Nichia) light source and an optical fibre coupled to an Ocean Optics USB2000 spectrograph for the in situ PL measurements. The reflectivity profiles of the as-cast films were recorded before they were sealed in jars charged with perdeuterated analyte under cotton wool to prevent solid analyte coming into contact with the films. The films were allowed to saturate with analyte for a minimum of 5 h before their reflectivity profiles were again recorded. To maintain saturation of the films with analyte throughout the neutron measurement the sample chamber was charged with cotton wool coated with the appropriate perdeuterated analyte. Analysis of the reflectivity profiles was performed using the Motofit reflectometry analysis program−6 Å−2 for the silicon substrate and included a native oxide layer on the surface of the substrate with an SLD of 3.47 × 10−6 Å−2. The errors in the modelled thickness, roughness and SLDs of the films given in Neutron reflectometry was performed using the Platypus time-of-flight neutron reflectometer and a cold neutron spectrum  and hydrogen peroxide ) before deposition of the dendrimer films by spin-coating from toluene solutions with a concentration of 10 mg mlHow to cite this article: Geng, Y. et al. Unambiguous detection of nitrated explosive vapours by fluorescence quenching of dendrimer films. Nat. Commun. 6:8240 doi: 10.1038/ncomms9240 (2015).Supplementary Figures 1-7, Supplementary Table 1, Supplementary Methods and Supplementary References


Electrofluidics is a versatile principle that can be used for high speed actuation of liquid interfaces. In most of the applications, the fundamental mechanism of electro-capillary instability plays a crucial role, yet it’s potential richness in confined fluidic layers has not been well addressed. Electrofluidic displays which are comprised of thin pixelated colored films in a range of architectures are excellent systems for studying such phenomena. In this study we show theoretically and experimentally that confinement leads to the generation of a cascade of voltage dependent modes as a result of the electro-capillary instability. In the course of reconciling theory with our experimental data we have observed a number of previously unreported phenomena such as a significant induction time  prior to film rupture as well as a rupture location not corresponding to the minimum electric field strength in the case of the standard convex water/oil interface used in working devices. These findings are broadly applicable to a wide range of switchable electrofluidic applications and devices having confined liquid films. Electrofluidics, also known as electrowetting, has its origins in electrocapillarity dating back to 18754710The device architecture described in this paper consists of a vast array of repeating electrofluidic cells, which in the case of an electrofluidic display serve as pixels1214Under application of a potential, the conducting liquid (in our case water) allows the surface charge to accumulate at the (oil-water) interface, which leads to an electrostatic compressive pressure across the gap or oil film. Provided that the potential is high enough an undulation in the oil film forms which will grow in time until the oil film ruptures. Counteracting the electrostatic pull on the oil film is a capillary restoring force121314121314In an unconstrained film one finds a continuous spectrum of waves depending on the film thickness and applied voltage18181819The theoretical part of the study reported here focuses on the underlying onset of instability in the oil film as opposed to late stageWe have conducted a high speed camera investigation of electrofluidic display pixels. We report here for the first time two key observations. Firstly the initial rupture of the oil film does not occur at the pixel wall, the precise location of film rupture depending on applied voltage. Secondly, we have noted an induction time of several milliseconds between the application of the voltage and the rupture of the oil film . The indAssume the reference state is a flat configuration of the oil film. Consider small deflections from the reference interface shape:The pressure acting on the film is composed of a capillary and an electrostatic component. In our system it is observed that the oil film thickness (5 μm) is much smaller compared to the observed undulation wavelengths (>50 μm). Therefore the analysis can be considerably simplified because under the large wavelength condition the lateral components of the electric field can be neglected. In most previous studies14151618C(h) is the capacitance of an oil film of thickness h, γ is the oil/water interfacial tension and V is the voltage applied to the system. The first term on the right hand side is a capillary restoring pressure while the second term is the electrostatic component of the pressure that tends to destabilize the film. The second derivative of the oil film capacitance gauges the way the electrostatic pressure is modulated as a result of infinitesimal changes in the height of the oil film12Positive values of this pressure promote film growth. P0 that occurs in the domain is there to ensure volume conservation. This method to deal with volume conservation has been adapted from18This expression  for the If the pressure is known for given change of the meniscus shape the fluid flow is governed by the lubrication approximation12It is justified to neglect the drag by the water in the upper layer since the water layer is much thicker so the drag is expected to be significantly lower compared to the thin oil film.Lx and the long dimension by Ly. Natural modes in this geometry are denoted by  and are described by the function:In this study square and rectangular cells are considered with lateral dimensions 150 × 150/150 × 315/67.5 × 315 microns and height dimensions defined by pixel wall structure 5.5/5.7/5.6 microns respectively. The short lateral dimension will be denoted by m and n uneven do not satisfy volume conservation, therefore in these cases considering pure modes is not justified. This shortcoming can be addressed by invoking the pressure jump P0, which gives rise to a small discrepancy compared to the pure mode analysis, which is discussed in These modes satisfy the pinning boundary conditions. However the modes with both The amplitude is governed by exponential growth (or shrinkage in the case of a negative time constant) with the following ratesThe electrostatic driving force is also written as a wavenumber:C''(h) ~ h−3, resulting in a constant growth rate. We do find however a strong dependence of the growth rate on applied voltage. Both these observations are at least qualitatively borne out by observation.Inserting the typical values for this system in k(V) > kmn. This marginally unstable energy criterion defines the voltage m, n] and  are defined at the point where the growth rate of mode  equals that of the neighboring mode . This dynamic criterion translates into: Now, from an energetic point of view positive growth rates can be realized whenever In systems exhibiting an instability without external forces, the fastest growing mode is the one that is selected.In the cells under study here, the wall is expected to exert an additional force on the interface close to it, as a result of fringe electric fields arising from a dielectric mismatch between the photoresist walls and the neighboring oil. In addition there may be other sources of inhomogeneity that can act as a trigger, such as over- or underfilling of a pixel or inhomogeneous pinning of the contact line by the photoresist. In case of sufficient external trigger we assume it is possible to have mode selection of marginally unstable modes.With this in mind we anticipate that the transition from one mode to the other should be found between the dynamic transition or at the marginally unstable point. To underline the theoretical universality of the transition we express the transition voltages normalized bywhich is a characteristic voltage of the system. For the capacitance the simple formula for a parallel plate capacitor is taken of the two layer system oil and fluoropolymer. This is justified if the water is sufficiently conducting such that the interface can be an equipotential one at the value of the applied potential.2 cells. In this case we have found the modes: /// and . The 150 × 150 μm2 cells showed modes / and   was used as insulator (thickness d ≈ 0.85 μm) material. The insulator was spin-coated on ITO/glass substrates with a resistivity of 100Ω Sq. The pixel walls were formed from an n-type photoresist  by lithography. As a result the fluoropolymer coated only the base of the pixels and not the pixel walls. The fluoropolymer surface was activated using a reactive ion etch (RIE) to facilitate the coating of the photoresist. Prior to liquid dosing the engineered substrate was heated above the Tg of the fluoropolymer (160 °C) to return it to its hydrophobic state. Oils are formulated by dissolving a non-polar dye in decane . A colored oil of 0.21 M dye concentration was used in this study. The polar phase sealed in the device was 0.001 M NaCl aqueous solution. The liquids were dosed to the substrate by raster filling at low speed (~1 mm/s). The cells were edge-sealed using a pressure sensitive adhesive (PSA) attached to the cover plate. The interfacial tension for the colored oil/water system was experimentally determined (at 25°) to be 0.020 +/− 0.002 N/m by the drop-weight method. The oil viscosity (at 25°) was found to be 1.23 +/− 0.01 mPa.s, measured by using a rheometer (HAAKE MARS).Amorphous fluoropolymer  with 1600 fps sample rate was employed for recording the electro-hydrodynamic instability phenomenon under an electric field. A series of square waves of 0.5 Hz frequency supplied by a waveform generator (Agilent 33500B Series) were applied to the oil/water system in our study.How to cite this article: Tang, B. et al. Interfacial electrofluidics in confined systems. Sci. Rep.6, 26593; doi: 10.1038/srep26593 (2016).