For this function class we present a runtime comparison between stochastic search algorithms using different search strategies. The (Escaping local optima is one of the major obstacles to function optimisation. Using the metaphor of a fitness landscape, local optima correspond to hills separated by fitness valleys that have to be overcome. We define a class of fitness valleys of tunable difficulty by considering their  Black box algorithms are general purpose optimisation tools typically used when no good problem specific algorithm is known for the problem at hand. No particular knowledge is required for their application and they have been reported to be surprisingly effective. Popular classes are evolutionary algorithms, ant colony optimisation and artificial immune systems. These examples fall into the family of bio-inspired heuristics, but there are many other black box algorithms, including Simulated Annealing or Tabu Search. While many successful applications of these algorithms have been described, it is still hard to decide in advance which algorithm is preferable for a given problem. An initial natural research topic towards understanding the capabilities of a given algorithm is to identify classes of problems that are easy or hard for it , 13, 38.Families of black box algorithms mainly differ in the way new solutions are generated (i.e. variation operators), how solutions are chosen for the next iterations (i.e. selection) and how many solutions are used by the heuristic in each iteration (i.e. population). Different variation operators, selection operators, population sizes and combinations of these lead to different algorithmic behaviours. In this paper we analyse the effects of mutation and selection in overcoming local optima.Two different approaches are commonly used by most black box algorithms. One strategy is to rely on variation operators such as mutation to produce new solutions of high fitness outside the basin of attraction of the local optimum. These are unary operators that construct a new candidate solution typically by flipping bits of an existing solution. Elitist algorithms (i.e. those that never discard the best found solution), mainly rely on such strategies when stuck on a local optimum. In a population-based algorithm different individuals may use different mutation rates to help escape local optima faster . Other vA very different approach is to attempt to escape by accepting solutions of lower fitness in the hope of eventually leaving the basin of attraction of the local optimum. This approach is the main driving force behind non-elitist algorithms. Compared to the amount of work on elitist black box algorithms, there are few theoretical works analysing the performance of non-elitism fitness valleys. These valleys can have arbitrary length d i.e., the difference in function values between the optima and the point of minimum fitness between them. More precisely, we define a valley on a Hamming path (a path of Hamming neighbours) to ensure that mutation has the same probability of going forward on the path as going backwards. The valley is composed of a slope of length In this paper we investigate this topic by considering the areas between consecutive local optima, which we call mentclass2pt{minimWe point out that understanding how to cross fitness valleys efficiently is a very important problem also in biology . From a We consider the simple elitist  introduced by Jägersküpper and Storch to give an example class of functions where a  .In this manuscript, we show that greater speed-ups can be achieved by SSWM on fitness valleys. Differently to the work in  where gleffective length of the valley.After presenting some Preliminaries, we build upon Gambler’s Ruin theory  in Sect.As a result, the runtime of the  function), and in the acceptance probability of these new solutions ( The (x) flips each bit independently with probability 1 / n. Conversely, SSWM and Metropolis analysed here use local mutations, hence the function mutate(x) flips a single bit chosen uniformly at random.In this paper we present a runtime comparison between the . Hence, Algorithm 1 takes a macro view to the adaptation process in that each iteration of the process models the appearance of a new mutation and its subsequent fate: either it is accepted with probability population size N refers to the biological SSWM regime , it can(Lemma 1 in ) For eveValley has slopes of The following lemma contains bounds on the expected duration of the game and winning probabilities for SSWM. Although (SSWM Gambler’s Ruin) Consider a Gambler’s Ruin problem as described in Theorem N for We start with the winning probability. Invoking Theorem Finally, for the expected duration of the game we call again Theorem mentclass2pt{minimWhile the optimisation time of the .The upper and lower bounds in Theorem Theorem n factor in the bounds  Consider a Gambler’s Ruin problem as described in Theorem Valley. Note that the required conditions are similar to those from Theorem Valley by simply applying Lastly, we make use of the previous lemma and the framework presented in Sect. Valley with The expected number of function evaluations ValleyPath consisting of m consecutive valleys of the same size. Each of the consecutive valleys is shifted such that the fitness at the beginning of each valley is the same as that at the end of the previous valley . It loosely resembles a “big valley” structure found in many real-world problems [ValleyPath is a very broad function class in that it allows for many shapes to emerge, from few deep valleys to many shallow ones. Our results hold for all valley paths with problems , 22, 30:Valley, instead of considering the whole Markov chain underlying ValleyPath we take a high-level view and consider the chain that describes transitions between neighbouring peaks. Since the peaks have increasing fitness, this chain is quite simple and allows for an easy application of drift arguments. By choosing the number of peaks to the right of the current peak as distance function, the next theorem shows that, if we can find constant bounds for the drift, we will only need to repeat the Valley experiment for as many peaks as there are in ValleyPath.As in the analysis for ValleyPath. Consider the points in time where the algorithm is on a peak and focus on transitions between different peaks. Let t-th time a different peak is reached. If the drift over peaks m is the number of valleys that compose ValleyPath, and Valley respectively.Consider any algorithm described by Algorithm 1 with local mutations on m peaks. The upper bound follows from the standard additive drift theorem [The lower bound is trivial since the algorithm can only move to a neighbour peak and has to visit  theorem , 18. \doValleyPath problem. We will also consider, apart from the peaks , the points of minimal fitness between them . For simplicity we will use the following notation.To compute the drift over the peaks ValleyPath Notation) Consider any algorithm described by Algorithm 1 with local mutations where the current search point is any extreme point (a maximum or minimum) of ValleyPath. If the algorithm is on a valley (minimum) we will denote by:If the algorithm is on a peak (a maximum) we will denote by:(4)(5)(6) first moving just one point towards the left slope and (2) winning a Gambler’s Ruin game starting with one dollar, using a notation in the same spirit as in the previous section . Note that, by applying Theorem In the next two theorems we apply the previous results on Valley (see Theorems The remaining conditions that Theorems ValleyPath with The expected number of function evaluations Valley to obtain the claimed result. Due to Lemma ValleyPath is shown for Metropolis in the following theorem.An equivalent result to that of the SSWM for ValleyPath with The expected number of function evaluations The proof follows exactly as the proof of Theorem entclass1pt{minimaNote that our approach can be extended to concatenations of valleys of different sizes, assuming We presented an analysis of randomised search heuristics for crossing fitness valleys where no mutational bias exists and thus the probability for moving forwards or backwards on the path depends only on the fitness difference between neighbouring search points. Our focus was to highlight characteristics of valleys where an elitist selection strategy should be preferred to a non-elitist one and vice versa. In particular, we compared the (


Size based separation and identification of particles in microfluidics through purely hydrodynamic means has gained significant interest due to a number of possible biomedical applications. Curved micro-channels, particularly spiral micro-channels with rectangular cross-section and the dynamics of particles in such channels have been extensively researched to achieve size based separation of particles. In this paper we present evidence that sheds new light on the dynamics of particles in such curved channels; that a unique particle slip velocity is associated with the focusing positions in the cross sections, which leads to a balance of forces. Our experiments therefore imply that the forces acting on the particle lead to convergence to an attractor in both the physical space (the cross section of the channel) and the slip velocity space. This phenomenon of inertial focusing has received much attention in the last decade in the context of microfluidics due to their potential biomedical applications such as for high through put identification or separation of cells. Curved microchannels, in particular spiral microchannels with a rectangular or square cross section have been extensively researched6 for this purpose. Particles are known to focus in a region of the cross section where the lift force and Dean drag balance each other and properly selecting the geometry of the channel and flow rates achieves controlled focusing of particles.The motion of spherical particles in a channel leads to preferential migration or focusing in a small region of the cross sectiondh is the hydraulic diameter of the channel and the dc is the diameter of the channel centerline. The relevant non dimensional parameters of the flow are the channel Reynolds number U is the average axial velocity of the fluid and v its kinematic viscosity. The secondary flow takes the form of two vortex cells, the so called Dean vortices, when the channel Reynolds number and the Dean number are small7. When small spherical particles of diameter dp are injected into a curved pipe they do not necessarily circulate in the cross section with the same velocity as the secondary flow. This difference in the cross sectional relative velocity causes the Dean vortices to exert a drag force on them called the Dean drag force, FD, in a direction perpendicular to the centerline. A second force on the particle in a direction perpendicular to the axial line is the lift force. The lift force is cause due to the curvature of the flow. Particles migrate to a positions where the lift force and Dean drag balance each other. These new equilibrium positions depend on the particle confinement ratio, 2.When fluid flows through a channel whose centerline is curved, a secondary flow arises within the cross section of the channel. The curvature ratio is defined as 10, in the last few decades have established that the lift force on a particle in a shear flow or a Poiseuille flow depends nearly linearly on the slip velocity of the particle in the axial direction. Particle slip velocity is defined as the relative axial velocity of the particle with respect to the undisturbed fluid axial velocity. Within the inertial microfluidics literature scaled relationships of the lift force have been developed that depend only on the position of the particle in the channel cross section, see for instance3. These scaling relationships ignore the factor of particle axial slip velocity entirely. In this paper we present evidence that the phenomenon of particle focusing is also influenced by the particle axial slip velocity. Unrelated to inertial particle focusing, it is known that neutrally buoyant particles lag the flow in a Poiseuille flow11. This is however the first work where we provide experimental evidence that particles have an axial slip velocity in a curved spiral channel. We show that distinct particle axial slip velocities are associated with particle spatial focusing positions in a spiral channel’s cross section, i.e. particle focusing phenomenon is due to a convergence of particle dynamics spatially as well as in slip velocity.Many theoretical studies, see for example16, have predicted the clustering and size based segregation of particles in some canonical two and three dimensional fluid flows. The calculations in these papers are based on a simplification of the Maxey-Riley equation17, that governs the motion of a small spherical neutrally buoyant particle,v is the velocity of the particle and u is the undisturbed velocity of the fluid at the same location, St is the particle Stokes number. The particle Stokes number is a measure of the inertia of the particle. It depends on the particle diameter dp and the fluid Reynolds number Re through the equation L is a length scale. The length scale relevant to particle focusing in a microchannel is the hydraulic diameter, dh, of the channel. So the particle Stokes number is r =  is determined by the equationw = v − u is the relative velocity (or slip velocity) of the particle. Equation  made of polyethelene with density 1 g/cm3 at a volume fraction of 0.03% was pumped through this microchannel at several constant flow rates until particle focusing was achieved. The small confinement ratio of the particles together with the small particle volume fraction ensures that the channel is not blocked by a particle or the velocity of the fluid in the channel is not significantly altered far away from the particle. The device is placed on a 3-axis flexure stage with differential micrometers that allow vertical motions of the stage with an accuracy of 1 μm. Figure A schematic of the top view of the microchannel device is shown in Fig. zf as the z-stage moves by a distance of Δzs are related by Snell’s Law. A light ray would travel through three media: air, PDMS and water, as shown in Fig. μ1 = 1 since the experiments were performed at a nearly constant temperature of 22 °C. The refractive index of PDMS is taken as μ2 = 1.4  and that of water is taken as μ3 = 1.33. The thickness of the PDMS slab above the channel is 1905 μm and the numerical aperture of the objective lens is 0.6. Calculations show that for this thickness of PDMS, Δzf = 1.56Δzs. We confirmed the values of the refractive index of the PDMS by fabricating several channels with different heights and calibrated the height traveled by the objective to the top of the device. We performed these measurements with and without fluid in the channel, to calibrate for the effects of the refractive index of water.We identified the screw gage reading of the differential micrometer for the baseline case of the channel bottom being at the focal plane of the objective. We did this by having markings on the glass slide before the PDMS device was bonded to the slide. Thus the zero height of the z-stage is calibrated. The change in height of the focal plane of the objective, Δ22. This is demonstrated in Fig. C1, shown in Fig. Particles that are pumped into the channel with a random distribution in the cross section with a random distribution of velocities eventually attain an equilibrium position with respect to the lateral walls of the channel. According to eq.  the convX magnification and a depth of focus of 0.9 μm, allowing one to discriminate between the quality of an image when the objective moves relative to the device by even a few micrometers. A vertical stack of images of particles in the spatial equilibrium positions is assembled as shown in Fig. i, j) in an image of size M × N is denoted by f. A sliding window of size a × b pixels centered around each pixel in the image is chosen. A sub image of the original image is first selected such that the sliding window can always fit at every pixel of the sub image. The indices of the pixels of this sub image are from M0 = a/2, Mf = M − a/2, N0 = b/2 and Nf = N − b/2. The average value of the pixel intensity in this window centered at  is denoted by p, q are pixel indices within this sliding window. The variance of these local variances is the global variance focal measure (GVFM) of the image23. Thusα is the total number of sliding windows. The GVFM calculations for our experiments are done through programs written in MATLAB. Images of particles in a vertical stack for which the GVFM is a local maxima have sharp well contrasted boundaries and correspond to the case where they are best focused23. It thus allows one to estimate with a very high degree of precision the vertical position of a particle in the channel. In Fig. μm. The graph has two maxima one at a height of 23 μm and the other at a height of 70 μm. These are the heights from the bottom of the channel where the particle images are the most well focused indicating that the particles focus at these heights. For reference the GVFM of a similar stack of images for the flow without any particles in shown in Fig. The heights of the particle equilibrium positions are measured from the bottom of the channel. To do this images are recorded at regular height intervals in the channel to observe how well focused the particles are in the images. The microscope objective lens (Nikon TU Plan Fluor EPI ELWD 50X) used in the height measurements has a 50A Photron FastCam SA4 camera recording at rates of 9000 to 13500 frames per second at a resolution of 512 × 512 was mounted on a Nikon microscope to record the motion of the particles. The velocity of the fluid in the channels in these experiments is of the order of 0.1 m/s. Since the velocity of the particles is of the same order as that of the fluid, the frame rate of the camera is sufficient to record changes in the position of the particles on the order of a few micrometers. The velocity of a particle at a focused position is measured by tracking the pixels of the center of the particle’s image in the consecutive frames. The identification and tracking of centers of spheres is done through MATLAB programs using the image processing toolbox.μm attain tightly focused equilibrium positions at about Re = 12.4. Once this particle focusing is attained the focal plane of the objective is located at the height where the GVFM has a maxima. This maxima occurs at two different heights in the channel, the first is at a height of 23 μm and the second is at a height of 70 μm from the bottom of the channel. Figure Particles of diameter 24 −4. An adaptive mesh was used with the maximum element size being 7 μm. The axial velocity of the fluid and velocity of the secondary flow were computed in COMSOL at the cross section C1  and ur − v at the right side (towards the channel center) of the particle. Due to the curvature of the velocity of the fluid shown in Fig. ul − v| > |ur − v|. Figure w = |ul − v| − |ur − v|. The values of ul and ur are obtained from a numerical simulation in COMSOL. The higher relative velocity of the fluid on the left leads to a lower pressure on the left than on the right side of the particle, producing a hydrodynamic lift force, Lf pointing away from the channel centerline and towards the channel inner wall. The hydrodynamic lift force increases with increasing values of Δw10. Figure w decreases very quickly for particles that are farther from the inner wall implying that the farther away a particle is from the wall, the lower is the hydrodynamic lift force on it. In the lower focusing window, the transverse velocity of the fluid (in the Dean flow) is towards the inner wall and a particle experiences a transverse drag that acts to push it towards the inner wall. A third force that acts on the particles in the cross sectional plane of the channel is the wall induced lift force, which always acts to push particles away from the wall. The wall induced lift force is strong close to the wall and decreases rapidly away from the wall24. The net force due to the Dean drag force and the hydrodynamic lift is balanced by a wall induced lift that pushes the particles towards the center of the channel. The wall induced lift force required to maintain this balance is smaller as a particle is farther away since the hydrodynamic lift force too decreases. Such a trend is observed in Fig. w decreases for particles farther away from the inner wall, which therefore experience a smaller inertial lift force.Focused particles in the curved microchannel have an  slip velocity. Particles which are injected at the inlet at an uncontrolled velocity converge to a narrow spatial region in the cross section of the channel and to a narrow range of slip velocities. In the lower focusing window a particle leads the flow. The velocity of the fluid relative to a particle in this window is w = |ul − v| − |ur − v|, is still positive for most of the particles as shown in Fig. In the focusing window located at about the channel mid height the particles lag the fluid such that the relative velocity of the fluid on the left side of the particles is in the opposite direction to that on the right side of the particle. However the difference, Δw would remain positive, i.e. relative velocity of the fluid on the side of the inner wall would still be smaller than the relative velocity of the fluid on the side of the outer wall. Therefore the direction of the hydrodynamic lift force would continue to point towards the inner wall as the analysis for the particles that lead the flow in the lower focusing window suggests.The relative velocities of the particles in the focusing window located at the mid height are negative and very small in magnitude compared to those of the particles in the lower focusing window. The sign of these relative velocities could be sensitive to errors in the experimental procedure in the determination of the height of the particles from the channel bottom as well as the subsequent processing of the images to calculate particle velocities. If due to such errors the particles are found to lead the flow with a small relative velocity, the sign of Δwy the difference in the relative velocity across the height of a particle in this window is even larger. This produces a vertical lift force pointing towards the wall that balances the wall repulsion which itself is very high due to the closeness of the wall. The large shear gradient induced lift force is possible due to the large relative velocity of the particles.There is a significant difference in the relative velocities of the particles in the focusing positions close to the bottom of the channel and the mid plane. The substantial relative velocity of the particle in the bottom focusing position is necessary to balance the vertical wall induced lift force. An analysis similar to the one illustrated in Fig. 25 in a straight channel. It is also shown in25 that this lift coefficient is such that the lift force has a large vertical component close to the bottom and top of the channels and a negligible horizontal component. This is the opposite close to the mid plane. While the overall lift coefficient is smaller at the mid plane, the horizontal component is the dominant part of it. It has been reasonably assumed in the literature that the qualitative behavior of the lift coefficient is the same in spiral channels. Our data actually indicates this to be so. The smaller relative velocity and smaller Δw is sufficient to balance the Stokes drag and the small wall induced repulsion at the mid plane height since the lift coefficient here has a mostly horizontal component. The larger relative velocity at the bottom focusing position is necessary since the lift coefficient has a dominant term in the vertical direction and negligible component in the horizontal direction. The large vertical lift force generated by the relative velocity balances the wall induced repulsion which is very strong close to the bottom surface.The lift force coefficient due to the shear gradient that multiplies the relative velocity varies with the location in the cross sectionWe did not observe a third focusing window close to the top of the channel, symmetrically above the lower focusing window, where a sufficiently large number of particles could be observed. We attribute this to two possible reasons. The first is that our technique of finding particle slip velocity is not refined enough to consistently and clearly image particles close to the upper surface of the channel,when only a few particles migrate to such a position. A second possible reason is that the particles are not perfectly neutrally buoyant, with many of them having a specific weight that exceeds 1 by a very small amount. The few particles that are perfectly neutrally buoyant could be focused in the upper window. This agrees with our observation of finding only a few particles in an upper focusing window. For heavier particles, in a hypothetical upper focusing position, the weight of the particle, the wall repulsion and the vertical component of the Dean drag force point away (downward) from the top wall. To balance these through the hydrodynamic lift force requires a larger positive relative velocity of the particles. Such a large slip velocity would not be allowed due to the Stokes drag force in the axial direction.28, where the experiments were conducted in a different regime of Reynolds-Dean number or channel geometry. For instance the experiments in28 used a channel with a high aspect rectangular cross section and small flow rates while26 used considerably higher Reynolds numbers (about 90–110). While it is understood through many experiments that channel Reynolds number, Dean number and channel cross section geometry effect particle focusing, our findings suggest another possible sensitivity to these parameters. At sufficiently high flow rates or very low flow rates , the slip velocity could in fact be very small. When channel Reynolds number is very high a significant inertial lift force could exist without a slip velocity, due to the inertia of the fluid29 since Δw is not zero even if w = 0. When the channel Reynolds number is very low, the lift force on the particle could be entirely dominated by the wall induced lift, leading to particle focusing positions close to the top and bottom channel walls. At the intermediate channel Reynolds number, investigated in this paper, the particle slip velocity plays a significant role. At the same time this makes a focusing window close to the upper wall less robust.Vertically symmetric focusing windows for particles are observed or have been suggested to exist in other work, see for instance11. It is demonstrated that particle focusing in curved channels depends on achieving an equilibrium in the spatial positions as well as the relative velocities of particles. The requirement of having an attractor in both the position and relative velocity space has an important bearing to several applications of inertial microfluidics. It may well be possible to refine the combination of active and passive methods that could take advantage of the slip axial velocity to focus or sort particles. For example a DEP (dielectrophoresis) force could impart small slip velocities to particles to shift or create equilibrium positions that otherwise may not exist in the flow. Essentially the lift force that scales linearly with the slip velocity can be changed without changing the flow using such techniques. The combination of such active-passive methods that harness the slip velocity dependent focusing could allow focusing and sorting to be performed in regimes of Reynolds or Dean number that is currently not possible.The observations presented in this paper demonstrate the influence of the relative velocity of particles on their equilibrium positions in a curved channel. Neutrally buoyant particles are shown to lead or lag the flow at different equilibrium positions, unlike in a Poiseuille flow where particles lag the flow


This data article presents a methodology and the corresponding code developed to perform and process stress relaxation tests where samples display superimposed (i) classical, continuous logarithmic relaxation together with (ii) sudden displacements manifest as abrupt stress decreases. The method extracts the activation area characteristic of the thermally activated mechanism that drives continuous plastic deformation in the material. We report stress relaxation data appertaining to as-cast (27) and annealed (2) aluminium microwires produced through a microcasting process. For an interpretation and discussion of the data on annealed microwires the reader is referred to “ The effect of size on the plastic deformation of annealed cast aluminium microwires”   The extracted values as well as the final retained datasets can be found in the ‘This dataset includes the datafiles (.txt) of the individual stress relaxations performed on aluminium microwires (99.99% and 99.999% purity) in one of two states: (i) as-cast state or (ii) annealed  LPM 200 load cell is used for microwires with During a stress relaxation test, the sample is loaded up to a pre-defined stress or strain level at which point the cross-head of the machine is stopped; the (decreasing) stress is then recorded versus time 2.2.2This method focuses on the continuous parts of the stress relaxations and calculates the apparent activation area based on this. The method implements a list of mathematical criteria that identify and omit the intermittent parts from the stress relaxations.The apparent activation area of a stress relaxation can be calculated using Eq. M the stiffness of the machine-wire system [MPa/s].With ® notebook (samplename_doublederivativemethod.nb) and its packages that can be found in the folder containing the stress relaxation data. The notebook can be used to extract the activation area from all stress relaxations of a single microwire at once.The binning method consists of 3 main steps: (1) binning the stress relaxation data to smoothen the time derivative, (2) discard bins that correspond to sudden displacement jumps and (3) calculate the apparent activation area based on the remaining bins. For each microwire these three steps are performed in the Mathematica2.2.3τσ or Δtotalτ/30, whichever is larger. The first criterion considers that the bin size needs to be larger than the standard deviation (measure of noise) on the stress signal (τσ). The second criterion limits the total number of bins to 30. In order to define a linear fit within each bin, the points need to be sequential in time. Therefore the size of the consecutive stress-defined bins  is used to define the final partitioning of the relaxation. An example of a binned dataset is given in the left-hand figure below  in the data divided by the bin length, it has little meaning).if 5.if 6.if To calculate the activation area of the continuous part of the relaxation, data pertaining to jumps (and the bins they are in) should not be included. To discard all bins containing a list of criteria was defined; the process is fully automated in the joined Mathematica® notebooks.The effect of the three last of these criteria on the bins used, and the corresponding remaining points in the dataset, is illustrated in 2.2.5•Calculation based on less than 3 bins.•The stress range covered by the remaining bins represents less than 20% of the overall stress decrease during the relaxation.•The calculated error is larger than the actual value.Although it is mathematically possible to calculate an activation area using the present (binning) method for every stress relaxation, this value is not always trustworthy, as the code will extract a value regardless of the extent or plausibility of data points. The following relaxations are deemed unfit to yield a valid activation area value:The final removal of stress relaxations inadequate to extract an apparent activation area is done in the joined excel file for each microwire.2.2.6During the derivation of the activation area from the time-stress data 2 linear regressions are performed, namely one to extract the time-derivative of the stress within each bin of a given relaxation and one to extract the apparent activation area from the logarithm of the derivative of τ for each relaxation. The activation area is inverted for plotting in the Haasen plot. Initial error on the signal will give rise to an error on the first regression . τ) at the calculated point is taken to be the average stress over the bin. The standard deviation of the estimated value of Additionally, because relaxation data are measured using a load cell, they will inherently suffer from white noise  τ plot to extract the activation area (a). This is a linear plot if continuous relaxation kinetics obey the often logarithmic stress decay law. Since we know the error on individual points on that plot, we can estimate their effect on the slope. In the present instance, one must take into account, in the linear regression, the fact that the error associated with each point is not the same (for the exact equations see p. 105 Eq. (6.12) of Ref. We now have a set of The error on the calculated slope will, thus, depend on the (variable) error on the fitted points:Δa as:The slope calculated in this manner is directly related to b²/Δa, for which the corresponding error σb²/Δa is given  as:Now, when plotting the information in a Haasen plot, we need to compute and plot .


Here we show that the determining factor for physical radiation enhancement effects for a clinically realistic cluster of heavy-atom bearing nanoparticles is the total number of heavy atoms packed into the cluster. We do this through a multiscale Monte Carlo approach which permits the consideration of radiation transport through clusters of millions of nanoparticles. The finding is in contrast to that predicted when isolated nanoparticles are considered and is a direct consequence of the Auger electrons playing less of a role for clusters compared to isolate nanoparticles. We further show that this result is agnostic to selection of the subcellular region considered to be sensitive to the effects of radiation, provided the inside the cluster of nanoparticles is not considered to be biologically active. Such nanoparticles are almost ubiquitously found clustered together in cells. Whilst in some cases there might be coating-dependent biologically based radiosensitization, a common feature to all such nanoparticles is physical dose enhancement able to act locally to produce large biological effects8. Although studies involving clusters of nanoparticles have begun to appear in the literature, these have either involved non-clinical beams9 or for periodic  arrays of nanoparticles10. Using a multi-scale Monte Carlo calculation formalism, we present calculations of physical enhancement due to clusters of many nanoparticles (millions in some cases), i.e. the form in which they are commonly observed.Nanoparticles with a core containing heavy-atoms and  a coating of lower average atomic number enhance effects of radiation during cancer therapy and have recently entered into clinical trialsHeavy atoms can be introduced into tumours either as nanoparticles or within small molecules. In order to study the physical enhancement effects, clusters comprised of three exemplar nanoagents were chosen - auranofin , a 1 nm radius core gold nanoparticle and a 25 nm radius core gold nanoparticle (AuNP). Both nanoparticles had a 1 nm thick citrate coating although this could equally well have been some other coating made from low atomic number elements.Dose distributions for clusters of nanoparticles were calculated as shown in Fig. The Calculations presented start with determination of the radial dose distribution for a single isolated nanoagent and proceed through determination of a series of intermediate quantities to result in predictions about the comparative effect placement and packing have on the physical radiation enhancement effects. Table 11 and their corresponding penetrations12. The smaller nanoagents’ RDDs extend to lower radii and correspondingly show larger dose depositions near their surface, even when viewed on a dose-per-primary basis.Radial distributions for isolated nanoagents are shown in Fig. Many copies of each nanoagent were considered to be randomly packed into a 1 µm diameter spherical cluster with a volumetric filling factor of 10%. The local energy enhancement, S1(r) was calculated:From S2(r) was calculated asSimilarly, These functions can be used to evaluate the enhancement to the number of lethal lesions/events for a wide range of scenarios. Furthermore, through examination many such clusters, the statistical spread in the effect across a large population of clusters was deduced, see Fig. L> since more of the nanoparticles interact with the radiation . The cluster-to-cluster variations in number of activations increase as the size of the nanoagent decreases - the decreasing number of activation events within the clusters of the smaller nanoagents leads to greater statistical variation across an ensemble. This effect is particularly pronounced inside a cluster, where activation of a single nanoparticle can give rise to a very big change in <L>, hence the particular large standard deviations observed for r < 500 nm.As the size of the nanoagent increases so does <S1(r) shows the importance of taking the RDD calculations out to large radii – although the RDD is rapidly falling off, this is largely due to the r−2 factor as the emitted electrons spread out over 4π steradians. This factor is negated when the total dose deposited in a spherical shell is considered.The slow fall-off of 13, the average number of lethal lesions induced can be written as:α and β are the usual parameters of the linear quadratic model, D is the dose and Vsens is the sensitive volume. I1 and I2 are given by:r) is a function describing the solid angle of a sphere of radius r centred on the nanoparticle cluster and lying inside the sensitive volume. Neglecting the effect of the angular variables implicit in these volume integrals is justified because the ensemble of clusters has no special orientational relationship with respect to either the incoming radiation beam or the biologically sensitive target.Within the framework of the local effect modelVsens. It is generally considered to be the cell nucleus although DNA damage and cell killing are not well correlated when AuNPs were used as dose enhancing agents14. In order to examine the effect of different choices for Vsens, the nucleus and the cytoplasm were separately considered. Since these clusters of nanoparticles are not generally found inside the cell nucleus16 only clusters outside the nucleus were considered. Clusters from outside the cell were considered as these could be in a separate cell. These scenarios and the values they produce for I1 and I2 are illustrated in Fig. These spherical clusters and cells provide a useful prototypic model with which to compare the number of lethal lesions/events of different nanoagents and different placements of the clusters in the cell. There is some ambiguity over what the biologically sensitive target is for cell killing, i.e. what exactly constitutes I1 is far bigger than I2, allowing I2 to be neglected in general17. Then, the dose enhancement factor (DEF) can be writtenThere is an open question as to whether the volume within the cluster but not occupied by nanoparticles can give rise to biological effect. In the main body of this paper, we consider this not to be the case. However, we discuss the alternative scenario in Supplementary Information. The results shown in Fig. I1, all of the factors on the right-hand side of this equation are dictated by the radiation biology pertaining to the particular clinical scenario – the only factor here which can be influenced for a therapeutic nanoagent is I1. For both choices of critical target, the curves of I1 for the different nanoagents are roughly parallel when seen on a logarithmic scale. This means that the associated DEFs will be in a roughly constant ratio, regardless of the placement of the clusters (comparing like-placement to like-placement). The departure from this parallel trend in Auranofin at larger values of d is due to the more rapid fall-off in its RDD  packed into the clusters. Comparing the 25 nm AuNP to auranofin at the same packing volume, the ratio of numbers of gold atoms is about 1.5 × 10−2. This value is close to the ratio of the integrals when most of the contribution comes from near the cluster. However, the value falls away from this value for larger values of d, a manifestation of the more rapid fall-off of auranofin’s RDD. This deviation is of little clinical relevance however since it concerns scenarios likely to lead to little clinical enhancement. These findings lead to the suggestion that the major factor in designing nanoagents for dose enhancement is their ability to deliver many high atomic number atoms within the clusters they form, or more generally, within the cell. This is because the extra ionisation events thus created far outweighs the effect of intra-nanoparticle energy attenuation.As is illustrated by Table This result follows directly from the fact that the 25 nm nanoparticle with a 1 nm coating has a higher fractional amount of gold than the 1 nm nanoparticle with a 1 nm coating. Since the total enhancement per cell is given by the enhancement per cluster multiplied by the number of clusters per cell (assuming like-for-like placements) the conclusion is that one would need roughly seven times as many clusters of 1 nm the nanoparticles to penetrate the cell as clusters composed of the 25 nm nanoparticles to have the same effect.The consideration of coatings is clearly important. If coatings were neglected then the enhancement per cluster would be the same for a given loading fraction, regardless of nanoparticle size.When comparing dose deposition outside of a cluster of nanoparticles to dose deposition outside of a single, isolated nanoparticle, it is clear that the lower energy Auger electrons provide a smaller contribution in the former case. Due to their small penetration, only a few of these low energy Auger electrons leave the cluster, these being associated with nanoparticles situated near the edge of the cluster. The diminished role of these low energy Auger electrons acts to negate the apparent benefit of the very large dose per ionization seen for the smaller nanoobjects in Fig. Given these factors, it is interesting to make a comparison between a spherical cluster of nanoparticles and a single homogenous sphere composed of elements in the same average proportions and the same total density as the cluster, i.e. to see what difference the fact that the cluster is composed of a nanoparticle + water inhomogeneous mixture rather than an equivalent homogenous sphere. The results of such a comparison are shown in Supplementary Figure rnp, to the edge of the cluster. That is to say the centres of the nanoparticles are all inside a sphere of radius I1 observed in Figure The inhomogeneous representation used with the superposition approach gives information about dose heterogeneity near the cluster as is seen in Fig. Of course, there are many simplifying assumptions being used to make these clusters of nanoparticles, such as that they pack with constant density, the coatings are rigid etc. One benefit of the inhomogeneous approach using superposition is that it allows these assumptions to be explored readily, something we anticipate will happen in future studies.et al. saw significant enhancement to DNA damage with 1.9 nm nanoparticles14 using very low X-ray energies. However, their modelling of the system suggests the effect will vanish at higher energies for these small nanoparticles, whilst it could be expected to persist for the larger ones. The conclusion from their model is corroborated by Jain et al.19 using 160 kVp photons when looking at both DNA damage and repair. Whereas, using 6 MeV clinical beams (which should make the observation of DSB enhancement harder) several reports of DSB enhancement have been presented: Chithrani et al. (50 nm diameter)18, Bebeco et al. (50 nm diameter)20 and Zhu et al. (20–74 nm)21. These observations are broadly in keeping with the trend shown in Fig. Although there is no definitive data set against which the result presented can be compared experimentally, there is plenty of corroborative evidence for the trend observed. Clonogenic survival studies have many potential confounding factors for instance, a poorly defined sensitive volume, so the discussion here is limited to observation of DNA double strand break (DSB) formation. McQuaid 8 but is a consequence of the proportionality of the DEF to I1 and the fact that each heavy atom present makes a roughly similar contribution  to I1, due to the fact that the large contributions to the RDD for very small distances have little effect when clinically realistic clusters are considered. Furthermore, this result is not dependent on the subcellular region considered to be sensitive to the effects of radiation, provided the inside the cluster of nanoparticles is not considered to be biologically active.A multiscale formalism, underpinned by local effect concepts, has been developed and used to describe the radiation enhancement effects due to clinically realistic clusters of nanoparticles. Using a simplified geometry of spherical cells containing spherical clusters of nanoparticles and considering thousands of such scenarios, sometimes containing millions of nanoparticles it has been possible to draw quite general conclusions relevant to the development of clinical radiation nanomedicines. The results indicate that the large dose enhancements found near small nanoparticles confer no practical advantage and is outweighed by their less efficient packing, unless there is a demonstrable and significant biological effect arising from energy deposition within the cluster. We wish to emphasise that this conclusion is not one which arises from previous studies considering isolated nanoparticleset al.7. In the second step, the particles (predominantly photons and electrons) from this spectrum were allowed to interact with a single nanoparticle. The outgoing phase space of all particles leaving the nanoparticle after any interaction was recorded, again in a manner similar to that of Lin et al. The fraction of particles undergoing interaction was also recorded in this step. In a third step these outgoing particles were transported through water with the dose deposition as a function of radius being recorded. This process gave rise to the RDDs shown in Fig. The calculations proceeded through 4 distinct phases. Firstly, a description of all the photons and electrons produced by a clinical linac at a depth of 3 cm into a patient was created. This description took the form of a phase space file which was then remapped to match the size of a single nanoparticle using the technique described by Lin et al. In the final step, the RDDs were used as kernel functions from which the dose distribution due to many nanoparticles was determined. This was done using the principle of superposition, with the kernel function being modified to take account of transport through both gold and water inside the clusters. For this step, a set of approximations were used which treated the elements comprising the cluster as a continuous mixture and which neglected the small amount of extra scattering brought about by the heavy atoms in the cluster. Through these approximations, a single RDD kernel could be used to determine the dose at any point due to every nanoparticle in a cluster undergoing an interaction with the incoming radiation beam. In this manner, thousands of calculations, each involving millions of nanoparticles in different clusters became tractable.Clusters of such nanoagents were constructed stochastically  software2 square field from the IAEA nuclear database25 was used to irradiate a water phantom of 20 by 20 by 40 cm3. In order to generate sufficient statistics, every entry in the database was used 10 times using the technique of particle recycling26. A cylindrical scorer of 2.5 cm radius was placed at 3 cm depth in the phantom to score all types of particles traversing its surface as is shown in Figure 7.In the first stage, a linac spectrum of 6 MeV TrueBeam Varian linac with a 10 cm27. The contracted phase space file representing the linac spectrum at 3 cm depth was recycled 4, 40 and 100 times for the irradiation of 25 and 1 nm GNPs and Auranofin respectively.The particles in those modified phase space files were used to irradiate 1 and 25 nm radius gold nanoparticles, each with a one nanometer citrate coating as is shown in Fig. 28 due to its high efficiency for the tracking of particles down to 100 eV in materials with Z <9929. In this stage, all physics interactions between irradiation and a single GNP were simulated, including the de-excitation processes that involves Fluorescence and the Auger cascades produced in the nanoparticle. All the secondary particles created inside the nanoparticle that were able to escape from it were recorded in a phase space file in ASCII format, which contained the energies, positions, momentum directions and weights of the particles, for subsequent use in the transport simulation used to determine the RDDs.The Penelope physics list in Geant4 was used to transport particles inside the nano-objects. The Penelope physics list has been widely applied for nanodosimetry30 was used to transport the electrons from the previous stage inside the water phantom. This physics list is able to track very low energy electrons in water down to a few 7 eV and has been widely validated for the tracking of particles at the nanoscale in water32. In this stage, all the electrons of the phase space file from the previous stage were released in a 1 cm radius spherical water phantom of as is shown in Figure The Geant4-DNA physics listTwo mathematically equivalent approaches were used, depending on the probability of a single nanoagent having more than one interaction with the radiation. When this probability was not negligible (25 nm radius AuNPs only in this study) all the AuNPs in the cluster were generated, ensuring the entire volume of each AuNP was inside the cluster and that none of them overlapped. For each nanoparticle, a number of ionisation events was generated stochastically using the Poisson probability distribution function with the mean number of events from the ionisation rate determined from the single nanoparticle simulations for AuNP and a WNP of the same size.This set was then pruned by discarding all those with zero ionization events. When the double-interaction probability was negligible (1 nm radius and Auranofin), only single ionization events need to be considered. This was done by generating the number of nanoobjects undergoing interaction  and their locations.The means to calculate the dose-deposition draws conceptually from the undergraduate physics problem of calculating the electric potential charges using the principle of superposition of charges. Accordingly, we borrow relevant terminology from the subject of electrostatics, referring to the point where we wish to calculate dose as the ‘field point’ and the centre of the nanoagent currently under consideration (i.e. producing the dose) as the ‘source point’. In this analogy the RDD takes the role of the Coulomb potential of a single charge. Due to the way the RDD is calculated, it already accounts for the attenuation of energy as the radiation passes through the water.ni is the number of ionisation events the ith nanoparticle has undergone. This result neglects the effect of other nanoparticles on the subsequent transport. There are two components to consider, scattering and energy deposition, as the radiation transports from the nanoparticle to the field point. We simply neglect scattering – this is justified because the system is quasi-spherically symmetric. Furthermore, Monte Carlo transport calculations explicitly considering the effect of a nanoparticle on the electron transport through a cluster  ratio of the stopping powers shown in Figure Energy lost due to transport through the cluster is accounted for by noticing that the electron particle is the main responsible for the depositing of energy in the medium when the nanoparticle is activated by gamma irradiation. Furthermore, the ratio of the stopping power for a water/gold mixture to the stopping power of water alone is constant to a very good approximation  maps from the distance between source point and field point to a new (slightly further) distance thereby accounting for the transport through the gold/water mixture in an average sense . Doing the calculation for ni set to the value for a 1 Gy fraction (to water) gives Making this approximation, the superposition equation becomes:g(r) was determined geometrically by taking a straight line from the source point to the field point and performing the following sum:rnp is the radius of the nanoparticle, lin is the distance travelled through the cluster but outside of the source nanoparticle but still inside the cluster and lout is the remaining distance to the source point. This formalism assumes a uniform probability density for finding gold anywhere inside the cluster but outside of the source nanoparticle. Strictly this is incorrect as the exclusion of nanoparticle centres within 2rnp of each other or within rnp of the edge of the cluster produces a slight suppression of probability in some regions of a given cluster, resulting in a small commensurate increase elsewhere. This approximation can be removed by taking the line integral of the ratio of stopping powers along the trajectory represented by lin. However, this more exact approach is computationally much more expensive and numerical simulation suggested the effect is negligible.S1(r) and S2(r) were calculated by numerical integration of equations r centred on the cluster. This was achieved by generating 1000 points on the surface of the sphere using Vogel’s method33, finding I1 and I2 can be calculated numerically from S1(r) and S2(r) provided Ω(r) is known. This function can be deduced analytically for either the cell nucleus or the cytoplasm from simple geometrical considerations. These analytic representations of Ω(r) were used along with interpolated representations of S1(r) and S2(r) to numerically determine the values of the integrals I1 and I2 shown in Fig. Due to the symmetry of the prototypical cells being simulated, the integral functions Computer codes used to perform these calculations will be made publicly available through the University of Manchester’s institutional repository upon its acceptance for publication.Supplementary Information


Nano-clusters are acclaimed to be very efficient absorbers of intense femto second light due to dominant collective mechanisms. Enhanced near 100% absorption due to collective linear plasma resonance compared to a small fraction of absorption by unclustered gas was an important drive in nano-plasma studies. Contrary to such perception, we show that if the pulse duration is (<100 fs), absorption is same irrespective of whether the systems are condensed to large clusters or not. So long as there are same number of similar ionizable systems in the focal volume, absorption is the same and such absorption can be accounted for by single particle response to the field and collisional ionization of atoms. Short pulse absorption by the single particle response can be comparable to the linear plasma resonance absorption for smaller clusters. Of these, the study with matter in the form of nanoclusters4 has its own special place. Having the unique features of solid-like local density confined to dimensions smaller than the laser wavelength and absence of the usual energy dissipation channels of bulk condensed media, allows one to study intense laser plasma physics with much lesser complexity. A combination of these properties result in high absorption5, highest charge densities in the plasma7, MeV ions with only keV electrons9, sub-ps bursts of hard x-rays10 and even acceleration of neutral atoms and negative ions12.The interaction of high intensity(>105. Current understanding of the interaction of intense laser pulse with clusters based on many important studies in this field can be summarised into the following steps: (a) Intense laser pulse ionizes atoms by field ionization  (b) Electrons released from atoms are driven by the laser field and contribute to collisional ionization. (c) Increase in electron density leads to formation of a plasma and collective absorption mechanisms like linear plasma resonance begins to enhance absorption of light. (d) Cluster expands/explodes both under hydrodynamic13 or Coulomb15 pressure finally releasing high energy ions and electrons. The first two steps contribute to the absorption of the laser light by single particle response to the external field and is referred to as the ‘non-collective’ absorption mechanism in this paper. The only major collective absorption mechanism we refer to in this paper is the linear plasma resonance mentioned in step-(c). Here resonant matching of the laser and local plasma frequency lead to boosting the local electric field and an enhanced energy transfer from the electromagnetic field to the plasma fields by increased charge separation and electon-ion collisions. This is referred to as collective because the local field inside the cluster is a function of the dielectric constant, a bulk property.A near 100% absorption when the Xe or Ar gas atoms is condensed to nano-clusters as compared to Ne or He that form very small or no-clusters has driven the nano-plasma science4, still many questions remain for want of answers. Are collective mechanisms important at all for nano-clusters with very short laser pulse interaction? Is absorption by an enhanced local field, a result of collective mechanism of the nanoplasma, the only way of large energy deposition in laser cluster interactions? Answer to many to these questions are often sought by computational models4. Particle-in-cell simulations or analytical computation schemes are used to switch on or switch off the collective mechanisms and the role of the laser energy absorption in the different steps is sought. In the many experiments performed in these systems, it appears that there is no clear strategy to experimentally differentiate absorption due to non-collective processes from collective absorption. We demonstrate in this paper, that a comparative study of laser absorption using Ar and N2 as target gases under the same conditions, provides a clear strategy to probe some of these important questions. We clearly establish that for 800 nm pulses shorter than 100 fs and having pulse energies of ~3.0 mJ focussed to intensities ~1016 W/cm2, absorption is predominantly by single particle response to the external field (steps a and b mentioned above). Absorption for short pulses is the same so long as the same number of atoms/molecules are present in the focal volume irrespective whether they are bound into van der Waals clusters or not. The short pulse absorption is dominated by the single particle response which itself can be as large as that due to linear plasma resonance. Only for longer pulse duration and larger clusters, does the effects of collective mechanisms have any significant contribution in influencing the absorption of energy  and ability of Ar to form bigger clusters makes little difference to the absorption. It appears that the absorption is only dependent on the number of atoms of Ar and N2 present in the focal volume. Since, the ionization potentials for the two gases are very similar (one reason why they were chosen in the first place) and the number of the atoms in the focal volume is only dependent on pressure, same absorption for such two gases can then be comprehended if the absorption is predominantly due to the single particle response to the external field with little influence of collective behaviour.Conventional wisdom in the nano-clusters studies covers pulse durations larger than 100 fs. On this timescales one would expect a larger absorption in a cluster aggregates2 has similar first few ionization energies. The first two ionization potentials for Ar are 15.75 eV and 27.69 eV and for N2 they are 15.58 eV and 27.9 eV, respectively16. Higher ionization potentials computed for N2 by the dissociation limits of the atomic ions are slightly higher than that of Ar. Ionization to these states are by barrier suppression or by field driven electron impact ionization. The thresholds for barrier suppression to a 7+ charge state for Ar is about 2.0 × 1016 Wcm−2 and it is 2.6 × 1016 Wcm−2 for N2. So it would make little difference in ionizability between N2 and Ar, especially for the intensities used here. Computation of the cross sections for electron impact ionization accounted by Lotz formalism17 show that the cross section for ionization of Ar and N2 for 1.2 keV electrons (ponderomotive energy at 2.0 × 1016 Wcm−2) differ between 15–20%. So if the absorption is due to field driven single particle response , Ar and N2 should be little different, irrespective of the fact that Ar is in the form of clusters while N2 is expected to only have a small fraction of much smaller clusters. Similar absorption with N2 and Ar shown in Fig. 2, unlike that in Fig. 2 reaching a saturation value. It can also be seen that as the pulse width is made larger, the absorption increases even more steeply and saturation is attained with even smaller size clusters or at a lower backing pressure. This and the absence of such a trend in N2 is indicative of collective mechanisms at play in this regime because while Ar forms larger clusters, the poorly aggregated N2 molecules do not allow for such processes and fail to reach the collective resonance leading to increased absorption.Ar and N2 as a continuous function of pulse width. This is presented in Fig. 2 clusters there is a systematic reduction in absorption with increase in pulse width. However, for Ar a different feature appears. For very short pulses (23–100 fs) the absorption is similar to that of N2. Absorption decreases till about 100 fs but after that there is a reversal in the trend and absorption increases again with increase in pulse width. Depending on the cluster size, the absorption increases to a peak value and further increase in pulse width decreases the absorption. The raise in absorption, peaking to large value depending on the cluster size and decrease further, is a well known feature of linear plasma resonance in clusters. This has been reported before with laser pulses of longer duration5 and is comprehended in many computation models18. It is attributed to the occurrence of a resonant coupling of the laser light with the collective oscillatory motion of the electrons having a characteristic natural frequency, the local plasma frequency. The field inside the cluster nanoplasmas that are much smaller in size as compared to the laser wavelength can be described as5ε and v are the plasma dielectric constant and the electron-ion collision frequency respectively and E0 is the laser electric field. With the internal electric field defined as above, the rate of energy deposition in the clusters, can according to the Poynting’s theorem, be written in terms of the rate of change of the energy density U as,ε and the plasma frequency ωp,3 gives a good exposition of this plasma resonance scheme. Owing to the much slower motion of the ions and the trapping of the quasi-free electrons in the potential of these background ions, the initial ionization of the cluster medium generates an almost solid density plasma with an ne close to 1023 cm−3 and thus a plasma frequency in the ultra-violet. But as the hydrodynamic forces expand the clusters over a few hundred fs(s) and the ne decreases to 3nc, where nc is the critical density for 800 nm, both the field inside the cluster nano-plasma as well as the rate of energy deposition reaches a maximum. This can be easily verified by plotting Eq. e/nc. This resonance clearly is a collective phenomena as indicated by the inclusion of collective properties like the dielectric constant in the equations above. The energy deposited due to these processes should also likewise be dependent on the size of the collective systems. Thus linear plasma resonance heating mainly due to the inverse Bremsstrahlung processes4 gives the maximum absorption.It should be kept in mind that the maxima of absorption due the linear plasma resonance and the resonance widths are strong functions of the laser intensity, wavelength and the cluster size. The physics issues brought out in this papers are limited to smaller clusters (<1/10 of the laser wavelength) and in the non-relativistic intensity regime (1015–17 Wcm−2). However if the cluster expansion is too slow, it is possible that the critical density is not reached before the end of the laser pulse and there is no resonance absorption. For smaller clusters, formed especially with N2, this resonance mechanism does not appear to affect the absorption significantly as it does for the larger Ar clusters at longer pulse duration. The expansion velocity in coulomb explosion depends on both the charge (average charge/atoms times the number of atoms/cluster) and mass of the atomic systems. Smaller cluster with smaller total Coulomb charge will expand slowly, even if the mean charge per atom is the same. For the same backing pressure, the energy of N is several times smaller than that of Ar.19. As a result, N2 clusters do not exhibit collective behaviour for the range of pulse duration studied here. This is clearly seen in the data presented in Fig. 2 while for Ar in Fig. In order to present the complete picture of the transition from one regime to another, a scan of the pulse durations was conducted to measure absorption in Ar and N20 could also be safely disregarded because of the similitude of the absorption values between the smaller N2 and larger Ar clusters as observed in this regime.A reversal and increase in absorption as the pulse duration is made shorter than 100 fs, not withstanding the changes in the intensity, is contrary to the expectations of linear plasma resonance absorption in clusters and has not been reported in contemporary literature. Short pulse absorption clearly depend only on the number of atoms present in the focal volume and can be attributed to the field dependent ionization of atoms, without the influence and modifications due to collective properties. Possibilities of anharmonic resonance due to increase in intensities10 Wcm−2, ionization is possible by tunnel ionization. The probability of ionization of an atom at a given position is computed using ADK  tunnel ionization which is given to be to be21ω is the atomic frequency, E is the laser electric field, and Ip is the ionization energy. At higher intensities, beyond the over barrier ionization threshold22 we use the direct ionization rate. The corresponding energy from the field is depleted so that in the next spatial slice of the laser pulse propagation is at lower laser intensity. The electrons generated at that point is assumed to be ponderomotively accelerated by the field and energy dispersed in collision ionization is calculated using the lotz formula17. The energy transferred to the atom by laser drive electron-impact ionization is again depleted. In the next spatial slice the laser pulse propagates with the corrected reduced intensity and similar computation is repeated. At the end of the propagation, the energy remaining in the field gives the unabsorbed light (It) transmitting out of the focal area. Absorbance is computed as (I0 − It)/(I0). Thus the total absorption of light is computed by integrating the fractional absorption through the different intensity regions and also accounting for loss in propagation through each cross-section.A computation of the absorption by field ionization is presented in Fig. 2 has small clusters is also incorporated by taking the appropriate electron density the clusters expand in time. To account for cluster expansion the hydrodynamic expansion rate has been considered with the average electron temperature taken to be about the ponderomotive energy.The experimental measurements for N2 at a pulse valve backing pressure of 40 bar is normalised with the computed absorption for the shortest pulse as shown in Fig. 2 can be effectively modelled as systems where the primary mode of light absorption is only due to the single particle - field responses: multi-photon ionization, over the barrier ionization and field driven electron impact ionization. We note that the computed absorption for pulse duration around 100 fs is smaller than that from experiments. The model that assumes only atomic ionisation processes does not include collective contributions like for example, there could be longitudinal drift of electrons as the laser propagates. For the experimental conditions used, there will be small N2 clusters and at larger pulse duration there will be some contribution to the absorption by collective mechanisms. The model that does not account for these effects is expectedly shows smaller absorptions than the experimental measurements. This difference is evidently much smaller in the very short and very long pulse durations where collective effects should not make much difference. Till about 100 fs where the absorption becomes minimal  absorption at 25 fs is same as it is at 250 fs. Larger the cluster, stronger is the absorption due to collective mechanisms. At 30 bar or <N>Ar ≃ 122, 500 atoms, absorption at 25 fs is 60% as compared to 75% at 700 fs. The shaded region in the Fig. 2 and Ar respectively.Also as is seen, for smaller clusters absorption due to collective mechanisms is comparable to the absorption due to single particle response. Figure 13. So models with non-uniform density are proposed such that the resonance condition is met for longer duration as different radial expansion zones come into resonance. It may be anticipated such a non-uniform expansion feature should also work for shorter pulse duration and there should have been signatures of resonance absorption even with shorter pulses. The experiments shown here does not support this possibility. For pulses shorter than the 100 fs and in the parameter space explored here, absorption is characteristically different and seem understandable well by single particle response to the external field. We note that the short pulses here in fact have higher intensity so the ability to produce stronger ionization is only larger. For clusters upto about 5 nm, absorption due to single particle response is comparable to that from the collective mechanisms in a direct experimental setting. We also note that recently theoretical simulations in the 5 fs pulse regime20 have demonstrated the possibility of frequency shifts in anharmonic plasma resonance absorption. The present experiments used longer pulse durations So, although the linear plasma response phenomenon is understood there are difficulties in comprehending the magnitude of absorption. An expanding nano-plasma passes the resonance point for a very short duration and power transferred to the cluster ensemble is not large enough to account for the x-ray/electron emission5. N2 with very similar ionizability as Ar works well as a system to compare the absorption due to collective and non-collective mechanisms.To summarize, intense ultrashort pulses impinging on matter are absorbed to start with by single particle response to field like multi-photon ionization, tunnel ionization or over the barrier ionization. Collisional ionization by the field driven electron further contribute to absorption by single particle response. Once the electron density increases and if the plasma frequency matches with the laser frequency, linear plasma resonance enhances the laser absorption. Our experiments show that in nano-clusters upto a size of about 14 nm, for pulse width shorter that 100 fs, absorption is predominantly due to the single particle response and there is a very little contribution to absorption by collective mechanism. Larger pulse duration with Ar clusters do show linear plasma resonance effect very similar and in concurrence to the previous experiments16 Wcm−2 intensity. The cluster jet is produced using a pulsed solenoid valve fitted with 500 μm conical nozzle. The mean size of the clusters in the beam is systematically varied by varying the backing pressure of the pulsed solenoid valve. The experiments are carried out in the regime where there is near 100% clusterisation with Ar and clusterisation is about 86% in case of N223. The cluster systems are characterised by Rayleigh scattering experiments carried out on this supersonic jet system earlier24.The mean cluster radius 〈r〉 and number of atoms per cluster 〈N〉 for the gases used Ar and N2 was estimated using the Hagena scaling parameter Γ*25. The pulsed valve is operated with a repetition rate of about 3 Hz and is synced to operate with the laser so that every gas pulse would interact with one laser pulse. We compare the signal level of the laser pulse that interacts with the cluster jet and those that does not, using a photo-diode placed after the region of interaction. The scattering losses are less than 5% as measured by a photodiode calibrated to a power meter. The average reduction in transmission as measured over The experiment is performed with an 800 nm 3.5 mJ laser having a 1 kHz pulse repetition rate and generates shortest pulses of 23 fs that are focused onto a cluster jet using an off-axis parabolic mirror to achieve 5 × 102 have very similar ionization potentials. Ar with larger polarizability forms large clusters even at low pressure in the backing chamber while on the other hand N2 has very low polarizability and does not efficiently form clusters. At any given backing pressure and identical operating conditions , the number of the atoms in the focal volume would be same except that in the case of Ar there would be nano-clusters and in the case of N2 it would be nascent molecules along with a smaller number of smaller clusters. Absorption of light in these two systems is the measure that we use to probe the role of collective mechanisms or the lack of it.The key strategy in this experiment is the choice of the gases made for comparison. Both Ar and N


We present an analysis of students’ responses to application-based questions on the topic of growth and control of microorganisms, from a questionnaire administered to 348 second and third year students of an Indian university who were enrolled in its undergraduate programs in Biotechnology or Microbiology. We examined aspects of the laboratory practice as reported by teachers and of the university assessment patterns that may explain our findings. Reports by teachers also included their views on the impact of the laboratory curriculum on building student capabilities. Studies such as this play an important role in informing the ongoing discourse in the country about much-needed reforms in undergraduate education.Our analysis revealed several lacunae in students’ understanding. Students’ performance on the questionnaire was also found to be poorly correlated with their academic achievement in the university examinations. Teachers’ reports revealed that there was a minimal student involvement in planning and designing of the experiments in their laboratory course; rather, cookbook protocols were commonly used by the students. There was a striking disparity between students’ stated career aspirations and their preparedness for them.Our analysis points to underlying issues in the teaching-learning and assessment process; we discuss these issues and possible alternatives to the current practices. This study, to the best of our knowledge, is the first in the country that has explored students’ conceptions for an elementary topic in biology education at the tertiary level. We believe that the results of the study will be useful in shaping the ongoing educational reforms in higher education and will also be useful in developing a concept inventory on this topic.The online version of this article (10.1186/s40594-018-0138-z) contains supplementary material, which is available to authorized users. The need for reforms in higher education in India has been widely acknowledged. In the recent past, several reports have summarized the state of higher education in India and emphasized the need for major reforms at both structural and functional levels Sen . Some reWe have recently embarked on a project designing interventions in the teaching-learning process that are practicable at the undergraduate level in India. As part of that project, we explored second year undergraduate students’ understanding of some basic concepts in biology—growth and control of microorganisms being one of them. Most experiments in microbiology require cultivation of microorganisms at one stage or another; students need to thoroughly understand the growth kinetics of a microbial culture to be able to quantify the microbial population for basic research and industrial applications. Indeed, the study of bacterial growth has long been considered an essential foundational concept in microbiology, one that all learners are required to master both in theory and practice Monod . Of equaWe report here our analysis of students’ responses to questions on the topic of growth and control of microorganisms. The study was extended to record teachers’ views on the curriculum and their general approach to laboratory sessions. We also report teachers’ opinions of the professions their students were competent to take up after the completion of the 3-year undergraduate degree course. We compare this with students’ own reports of their career aspirations.The major research question that guided this study was:What conceptions do undergraduate biology students have regarding the growth and control of microorganisms?What are the curricular issues, if any, that might be impeding the development of conceptual clarity and higher order thinking skills in this topic among students?(a) Are students’ career aspirations aligned with their conceptual understanding of the topic?(b) What are the teachers’ perspectives on the skills and capabilities that the undergraduate course is developing in students?The subsequent extension of the study was conducted to answer these research questions:To the best of our knowledge, this is the first discipline-specific study in the country that has investigated student ideas about this foundational concept. We believe that our study can provide insights for improving teaching-learning for this topic, thus contributing towards undergraduate biology education reforms in the country. The insights into alternative conceptions held by students may also be useful for building tools such as concept inventories. A recently developed concept inventory in microbiology , and the second group consisted of 52 students in the third year (TY) of their degree course. These students, typically aged 19–21 years, came from five colleges, affiliated to the same university, following the same curriculum, and were enrolled in a 3-year degree program in Biotechnology (BT) or Microbiology (MB). These colleges were selected because they spanned a range of minimum eligibility scores (Table Eligibility to enroll in the undergraduate program is based on the grade 12 scores in examinations conducted by boards of education across the country. Copies of mark sheets for grade 12 and the first year of B.Sc. were obtained from students and the minimum eligibility score for admission from the college authorities. For more details on the distribution of participants’ grade 12 and first year marks (in their discipline—MB or BT), see Additional file Due to logistical constraints such as college schedules, the study was spread over nearly a year. Since the questionnaire was administered to college V towards the end, the second year B.Sc. students from that college had already graduated to the third year of B.Sc. at the time of administering the questionnaire. So, in addition to the second year students, it was administered to the third year students as well. Our motivation to include them was also that they had undertaken other advanced courses such as Industrial Microbiology/Biotechnology , where sterilization of media/equipment and growth of microorganisms are important base concepts. Third year students’ responses could act as an indicator of improvement in understanding, if any, after having revisited the same concepts.Two questionnaires were administered to all students (group 1 + group 2)—the first questionnaire checked their level of conceptual clarity and included a section on feedback on the difficulty level and the language of the questions  and 15 min for the feedback on the questionnaire. The feedback on the questionnaire was checked for internal consistency. Those found to be internally inconsistent were not considered for the analysis, for example, a student reporting that the language of the question was clear and precise and also marking that the language was confusing for the same question. The second questionnaire, to collect data on students’ career aspirations, was administered online to all 348 students; a total of 192 responses were received.The third group consisted of 7 in-service teachers, who had been teaching these courses to undergraduate students for at least 10 years. A feedback on the expected challenge level and language of the questions was obtained from them online.The fourth group consisted of 55 in-service teachers  having a range of experience (1–34 years) of teaching undergraduate students; some of them had taught the student participants of our study. These 55 teachers were administered an online questionnaire seeking their perspectives on the current laboratory curriculum as well as the practices followed by them in negotiating the curriculum. Students and teachers volunteered their participation in the study, and we did not incentivize or compensate them in any way.The questionnaire developed for the study consisted of both open-ended and multiple-choice questions on three topics—the eukaryotic cell membrane, evolution, and growth and control of microorganisms. The questions were designed from material common to both Microbiology and Biotechnology courses and covered in their syllabus up to the first year of B.Sc. The questions for each topic were designed to cover more than one concept and test higher order thinking skills. None of the questions asked for mere recall; students were required to justify their answers even in the multiple-choice questions.The questions were then assigned an expected challenge level for students—easy, moderate, or difficult. This was guided by our expectations based on how often students had encountered and/or were required to apply the concepts during their coursework. Their familiarity with the format of the question was also taken into account Table . FinallyThe researchers independently analyzed and scored responses received from the students of college II to the questionnaire. Minor discrepancies (< 0.5%) were resolved through discussions among the researchers and the expert. The final category and score for each question were arrived at by consensus. The questionnaire was then administered to the other four colleges. It was found that the categories of responses that emerged during analysis of responses from college II were exhaustive: Responses from all colleges could be easily be assigned to these categories. The scoring and coding scheme for the three questions is presented in Table A correlation analysis was performed with the students’ scores in our questionnaire and marks scored in the first year college exams using LibreOffice Calc (available in the LibreOffice software package of a Linux system). The first year college marks used for this study were summative of class assignments, internal college exams, and final university exams for MB/BT courses as has been stated in their first year transcripts. Further, we examined the nature of questions appearing in their routine assessments by categorizing them based on revised Bloom’s taxonomy, as described by Krathwohl . A totalThe range of student responses to just three questions on growth and control of microorganisms brought out several lacunae in their understanding of the topic. The percentage of responses of second year students in each category for the three questions is presented in Fig. A major lacuna that became evident in students’ understanding was that they were unable to differentiate between two very distinct processes—sterilization and disinfection. Student responses to two parts in the “easy” question, and the “difficult” question indicates that these two processes were understood to be identical by them. For example, in the former, students responded that contaminated hospital linen could be sterilized using detergents or a disinfectant solution. Curiously, some students also suggested the use of refrigeration as a technique for sterilizing vitamins, perhaps confusing it with preservation. In case of Petri dishes, a variety of suggestions were obtained—using disinfectants such as sodium hypochlorite, phenols, alcohol, washing with detergent, boiling, keeping in an incubator, or increasing the pH. In the latter question, students were explicitly asked to comment on the situation . The incorrect responses were those that equated disinfection (the procedure followed by the doctor) to sterilization (the procedure that should have been followed by him). A smaller fraction (3%) in this category, in fact, commented only on the use of soap—that it should not have been used as it is an “irritant.”In suggesting a method of sterilization for contaminated hospital linen and a vitamin solution, students did not take into account whether the method would be suitable for sterilizing the given material. For example, students proposed the use of UV radiation or laminar flow, clearly not considering the bulk of the material or that the entire surface area of the linen may not get exposed to UV light. In case of vitamin solution, 46.3% of respondents failed to acknowledge the heat labile nature of vitamins and suggested the use of a hot air oven or an autoclave. It was intriguing to note that a smaller fraction (7.8%) of students who did recognize the heat labile nature of vitamins still proposed incorrect methods. Some advocated against the use of a hot air oven because of its high temperatures (between 160 and 200°C). Instead, they recommended that the solution be autoclaved or heated at lower temperatures (between 50 and 60 °C) in order to prevent degradation of vitamins, failing to take into account that high temperature (121.6°C) exposure during autoclaving will also lead to degradation of vitamins or that sterilization will not occur at lower temperatures.A large fraction of students’ responses to the subparts of the “easy” question was partially correct (the method was written without a justification). Such results suggest that students appear to be familiar with many methods used for sterilization but are only able to recall the names without actually understanding the principle of the technique.Students responded to the “easy” question by suggesting, irrespective of the material to be sterilized, some commonly used methods such as autoclaving, using a hot air oven, or a laminar airflow hood—equipment that has a conspicuous presence in an undergraduate laboratory. They seem to be less sure about methods such as membrane filtration, which, although taught in theory classes, is not a commonly used technique for sterilization in their laboratories.The objective of designing this question was to gauge whether students are able to acknowledge the exponential growth of the microbial population and are able to calculate the total cell count after a given period of time. Only 14.5% of the students were able to calculate the final cell count correctly and another 10% ) were at least able to recognize the exponential growth pattern. It was noted that almost a quarter of the students considered the microbial growth pattern to be linear while almost half of the students did random calculations unrelated to the question. This is particularly concerning because students were expected to be familiar with microbial growth kinetics which is dealt with in their curriculum in detail, both during the laboratory (five 3 h sessions), and theory (15 lectures for 50 min each) classes  and 7 teachers (group 3). We refer to the students’ responses as “rating of the difficulty” while the rating by teachers and researchers is referred to as the “expected challenge level” for reasons that are self-explanatory. Second year students’ rating of the difficulty is shown in Table For the question marked “easy” by the researchers, a majority of students, as well as the teachers, have marked this question as easy or moderately difficult; more or less in alignment with the challenge level assigned by the researchers. One reason why this question could have been considered to be of moderate difficulty by some students and 5 out of 7 teachers is that the question did not just ask for a mere recall of the technique but also demanded an explanation for suggesting that technique. This would require a thorough understanding of the principles behind the techniques.n = 255), on the other hand, gave a mixed response to this question .Unlike the researchers, all the 7 teachers considered the problem on the growth curve to be “difficult” from the point of view of the second year students, stating that “students find any question where some application of knowledge is required, difficult.” This finds corroboration in the student responses, where a large percentage (23% + 40.2%) of them answered this question incorrectly owing to errors in basic mathematical calculations and gaps in the understanding of the growth curve. This indicates that the teachers did identify that students face difficulties in the application of basic quantitative skills to biological concepts and hence categorized the question as “difficult”. Intriguingly, students  marked this question to be difficult/very difficult while the rest found it easy or moderate. This is generally consistent with their responses to the question where 66% of students gave either a correct (21%) or partially correct (45%) response. Five of the teachers marked the question to be moderate/difficult and reported that students would find it difficult to answer questions which required the application of concepts and suggested that we “pose a proper question, like “do you think this is the correct practice? Justify.” Students will not know what is expected of them.”expected to know after having studied these topics in their classes.The slight difference between the teachers’ and researchers’ categorization of questions could be attributed to, perhaps, the difference in perspectives. The teachers have marked the category based on their knowledge about what students do know whereas the researchers have categorized the questions based on what students in their second or third year would be An analysis of students’ performance in our test led us to explore the potential reasons for lacunae in their conceptual understanding of growth and control of microorganisms.R) was found to be 0.34 (determination coefficient (R2) of 0.119), indicating a weak correlation between their scholastic achievement and the marks obtained on our questions as depicted in Fig. A correlation analysis was performed between group 1 students’ marks on questions on growth and control of microorganisms in our questionnaire and their scores in the first year University exams in Microbiology (MB) and Biotechnology (BT) courses. The correlation coefficient , as many as 14 questions were clearly categorized under “remember” (R) and 6 under “understand’” (U) by all evaluators  to that part of the questionnaire that required marking on a Likert scale. Open-ended questions from the questionnaire are being discussed in the text. An analysis of teachers’ responses to our questionnaire probing teaching practices revealed that students have little involvement in the design of or in the preparation of laboratory material for experiments. For instance, as part of their experiments, students are routinely required to grow microorganisms in the laboratory which require an application of sterilization methods. In response to the question asking teachers to list the role(s) and contribution(s) of the laboratory staff, students, and teachers in planning and preparation of the practical sessions, all teachers reported that preparation and sterilization of material needed for the experiments are done by laboratory staff; the role of the students was listed largely as “performing the experiment”.Table An overwhelming majority (about 80%) of teachers have also reported that instances of students consulting research articles or books for suggesting experiment protocols were rare. This was particularly surprising because a large fraction (69%) of teachers reported that they always/very often encourage students to read research articles related to the experiment. Also, when asked to opine on the statement that “students do not understand research articles”, a substantial fraction (47.3%) disagreed with the statement. The reason for students not referring to protocols presented in research articles can be inferred from teachers’ responses where a large section (82.7%) has reported that students are always, or at least very often, given the protocol to be followed in a particular laboratory session. A substantial fraction (65.2%) of teachers also agreed that these protocols are the ones printed in their laboratory journal (handbook). Laboratory journals generally contain protocols that have been standardized to obtain a particular result and do not allow for any variation. Based on these findings, we surmise that teachers encourage students to read articles as secondary sources of information and not for developing a hypothesis or for designing experiments.The questionnaire also consisted of a section probing the level of positive impact of the laboratory curriculum on some aspects of student learning. The teachers reported that the laboratory curriculum has a high/moderately positive impact in building a link between theory and laboratory (81.8%). Consistent with this response, teachers (69%) agreed that they conduct laboratory sessions in alignment with what students are studying in the theory classes. They also reported a high/medium positive impact on the learning of technical skills (96.4%), allowing for student engagement with the principles of the methods they are employing in the experiment (100%), attracting students to research (58.2%), providing with opportunities to develop quantitative skills (58%) and analytical skills (56.4%), and motivating students to pursue science for further education or careers (80%). However, the level of positive impact was, largely, reported by them as moderate or that they were unsure of the impact in aspects such as providing an understanding of research methodology (67.3%) and making students competent to deal with unexpected situations (69.1%).In summary, the teachers are of the view that the laboratory curriculum and their practices satisfactorily aid in student learning. However, this view is not supported by the findings of our study.Among the 192 students who responded to the questionnaire on career aspirations, the most favored career options were research/academics (24%) or employment in an industry (26%) (food industry being a favorite) Fig. . A smallThe analysis of students’ conceptions of growth and control of microorganisms revealed several gaps in their understanding of this basic topic. A similar performance by third year students is further indicative of a systemic failure—in both teaching and assessment—to build even the basic practicable knowledge expected of an undergraduate student enrolled for a degree in microbiology or biotechnology. The results corroborate the finding of a previous study by Phadnis and Pandit  that theAnother area of concern is the rather dismal performance in the question requiring quantitative reasoning; indeed, biology students’ problems with mathematics are well documented  of students’ performance was consistent with our assessment of students’ conceptual understanding. Their evaluation of students’ preparedness for careers in these fields is also consistent with our findings. We note, however, that there was a striking mismatch between students’ stated career aspirations and their preparedness for them. Yet the curriculum, particularly aspects related to laboratory practice, does not seem to address these issues.Our findings on students’ conceptions, together with teachers’ responses on the laboratory curriculum and evidence for inadequacies in the assessment system, highlight the need for reforms in the teaching-learning process in the undergraduate biology courses for creating a meaningful learning experience for students. This would be the first step in helping students to achieve competence for careers they aspire to pursue. We have also pointed to feasible changes in the curriculum which can aid both curriculum designers and in-service teachers in improving the current practices. We hope we have demonstrated the value of focused discipline-based education research in pointing out specific measures towards reforms in higher education.Additional file 1:Figure S1. Distribution of students’ marks in (a) grade 12 and (b) first-year (MB/BT). (EPS 45 kb)Additional file 2:Questionnaire to seek students’ rating of the difficulty of content and language of the questions on growth and control of microorganisms (appearing after every question). (DOCX 23 kb)Additional file 3:Table S1. Categorization of questions based on revised Bloom’s taxonomy. (PDF 185 kb)Additional file 4:Table S2. Syllabus for the topic of growth and control of microorganisms over the 3-year degree course. (PDF 39 kb)