charging of grains of identical insulating 
 materials during collisions 
 has been of considerable interest recently , both for its intrinsic 
 physics and for its applications to situations ranging from volcanic 
 dust plumes and desert sandstorms to industrial powder processing .
however , work up to now has not explained why in some instances 
 charging grows , rather than diminishes as one might naively expect , 
 and can run away extremely rapidly , leading to electrical discharges : 
 lightning .
outstanding examples of runaway collisional charging involve 
 ice , in thunderstorms both on earth and 
 on other planets , and , it is speculated , in the 
 solar nebula ; these instances concern granular media whose particles 
 also undergo nucleation , growth , and fission , so that they , in effect , 
 reproduce . in this work ,
we demonstrate with a minimal dynamical model 
 that secondary nucleation , production of a new particle from an existing 
 particle , is a key process in producing runaway electrical charging 
 in self - replicating granular matter .
we concomitantly present the 
 results of experiments on growing ice in situ from the vapor phase 
 within an environmental scanning electron microscope .
we show that 
 an effect of the electric field is to induce the formation of fast - growing 
 ice  palms 
intermediate in morphology between whiskers 
 and dendrites ; the ease of breakage of these palmlike formations will 
 clearly favor secondary nucleation and hence runaway electrification .  in a thousand seconds , more or less , its volume increases 
 a thousandfold , the intensity of its electric fields increases a thousandfold , 
 and its electric energy increases a billionfold  ; thus described 
 vonnegut the tremendous metamorphosis a cumulus cloud undergoes to 
 become a cumulonimbus or thundercloud .
thundercloud electrification is a consequence of ice particles colliding 
 within a cloud and exchanging electrical charge .
charge dipole development 
 in a thunderstorm is due to the physical separation of particles with 
 opposite charges inside the cloud : larger , heavier particles will 
 fall , while smaller , lighter particles will rise in the updraft , owing 
 to their different dynamics , and these 
 particles carry different charges .
usually , 
 in thunderstorms , the smaller ice particle is an ice crystal and carries 
 positive charge aloft in the updraft ; the larger graupel ice particle 
 falls with an opposite negative charge , leading to a typical thunderstorm . 

we may contrast the foregoing with charging in other granular media , where it has been argued that simple geometry , without growth and 
 fission processes , leads to a net transfer of electrons from larger 
 to smaller particles , so that smaller 
 particles tend to charge negatively and larger ones positively . whether 
 this differential charging tendency operates one way or the other 
 depends on the microphysics , which differs for different materials , 
 so this polarity differs .
for our present purposes , however , what 
 is relevant is that there should exist such a triboelectric charging 
 tendency in one sense or the other .
here we build a minimal model 
 ( figure 1 ) incorporating solely the collisional 
 dynamics of charge transfer plus nucleation , growth , and fission processes , 
 which we aim to have general relevance to self - replicating granular 
 media .
anatomy of our minimal model of charging of friable self - replicating 
 granular matter through secondary nucleation , incorporating the processes 
 of particle growth ( ri  ri + 1 ) , advection 
 ( with speed ui ) , collision , 
 charge transfer , and fission ( with probability s ) .
we consider ( figure 1 ) a one - dimensional 
 system of length l , within which we place randomly n neutrally charged particles , qi = 0 for i = 1 , ... , n , of size ri extracted 
 from a gaussian distribution of sizes with mean r = 1 and standard deviation . these particles grow at a constant 
 rate , independent of particle size , that sets the time scale of the 
 problem . at the same time , they sediment in an upward flow of constant 
 magnitude at their terminal velocity , ui , determined by the instantaneous balance of fluid 
 drag and gravitational forces .
the sedimentation speed is normalized 
 by the updraft speed ( i.e. , ui = 1 for passive tracers ) and is approximated by a linear function 
 of the particle size ri , ui = 1  ( ri / rc ) .
this function is positive ( negative ) for ri below ( above ) a critical value rc .
the sedimentation speed of all particles in the simulation 
 is updated every integration time step ; in this way , small particles 
 that are initially advected upward by the updraft slow their upward 
 motion as they grow and begin moving downward after reaching the threshold rc . when the trajectories of two particles i and j cross , they collide . to overcome 
 the 1d limitation of 
 the model ,
we allow particles to get past one another after the collision , 
 so that after each time step particles end up in whatever final position 
 their velocities prescribe .
both mass and charge are conserved during 
 a collision but , while charge is transferred in every collision ( the 
 smaller particle leaving the collision positively charged : qi  qi + 1 and qj  qj  1 with ri < rj ) ,
we assign 
 a certain probability s for fission to occur for 
 each of the two particles involved in a collision .
if fission does 
 occur for particle i , a new , neutrally charged particle 
 with radius 1 splits from it , reducing its size to ri = ri  1 .
although fracture of particles has been shown 
 to lead to additional charging , we left 
 this effect out of the model for the sake of simplicity as it does 
 not change qualitatively the presented results .
boundary conditions 
 are absorbing : particles leaving the system 
 through the upper or lower boundary are absorbed there , and do not 
 participate further in the dynamics , but their charge accumulates 
 to the total charge at the boundaries : qu(l ) = qi for i leaving the system through the upper ( lower ) boundary , 
 respectively .
the total large - scale charge separation is calculated 
 as q = qu  ql . for given initial conditions ( set completely 
 by the concentration 
 of particles , 
= n / l , and 
 the initial spread of the size distribution ,  ) , the behavior 
 of our minimal model then depends on just two parameters , the critical 
 radius rc and the secondary nucleation 
 rate s. in what follows , we set rc = 8 ,  = 2 and  = 0.1 and explore the behavior 
 of the model with respect to the secondary nucleation rate s. we note that electric forces are not considered in the 
 model but implicitly ( through the secondary nucleation rate responsible 
 for the process of fission ) .
figure 2 displays the typical transient 
 dynamics of the charge distribution in the model .
the figure shows 
 an initial stage in the transient dynamics chosen to exemplify how 
 the charge separation process operates in the model .
as time passes , 
 as indicated by black arrows , light positively charged particles move 
 toward the upper boundary while heavy negatively charged ones move 
 in the opposite direction , contributing to a non - negligible dipolar 
 large scale charge separation q .
regions where 
 localized collisions are produced generate large charges that are 
 then separated by differential advection . even at those early stages 
 in the process , the global dipolar structure of the system , quantified 
 by the large - scale charge difference between the boundaries q , is the dominant field .
figure 3 shows particle size and charge distributions for the upper and lower 
 boundaries .
as time passes , light positively charged particles move toward the 
 upper boundary while heavy negatively charged ones move in the opposite 
 direction , as indicated by arrows , contributing to a non - negligible 
 large - scale dipolar charge distribution q . 

particle size ( rj ) 
 and charge ( qj ) distributions 
 for the system upper boundary ( left panels ) and lower boundary ( right 
 panels ) .
the total charge separation 
 q at a finite time step ( which is a measure 
 of the speed of the charge - separation process ) displays nonmonotonic 
 discontinuous behavior as a function of the secondary nucleation rate s. below a critical value sc , the long - term dynamics reaches an emptying state with no particles 
 left in the domain and a finite ( and small ) final charge separation ; 
 as there are few secondary nuclei , the initial particles grow and 
 collide very little before leaving the system .
occasionally some charge 
 is produced , but not much . above sc , the 
 number of collisions grows , and with it the charge transfer .
as the 
 charge acquired by the particles is correlated with their size , charge 
 separation occurs too .
the system approaches a steady state with an 
 average nonzero number of particles in the domain and an exponentially 
 divergent charge separation , as shown in the inset of figure 4 .
this exponential charge growth is limited in nature 
 by electrical discharges ,  lightning  .
the total charge separation q displays nonmonotonic 
 behavior as a function of the secondary nucleation rate s. inset : time evolution of q for s = 0.1 shows the exponential growth characteristic of all 
 values of s > sc .
however , for secondary - nucleation 
 rates much greater than sc , there are 
 more and more collisions .
the particles 
 undergo more fission into secondary nuclei , and although much charge 
 is produced , it does not separate as well ; charge separation is still 
 exponentially divergent , but the growth rate decreases beyond sc .
there is thus an intermediate optimal value 
 of the secondary nucleation rate to produce the greatest charge separation . 

this critical value sc  1/(rc ) is 0.0625 in the simulations ( dashed line 
 in figure 4 ) .
a smaller value of rc ( which means a weaker updraft ) requires secondary nucleation 
 to occur more often ( larger sc ) in order 
 to ensure some particles are advected upward ( leading to charge separation ) , 
 while the initial density of particles ,  , controls the collision 
 probability .
it is worth mentioning that even in the limit of very 
 high particle density  , in which many collisions occur at early 
 stages , charge separation is minimal in the absence of secondary nucleation . 

earlier work 
 of ours made us suspect that secondary nucleation 
 ought to be important in runaway collisional charging .
previously 
 we have shown that the nonlinear feedback effects of secondary nucleation 
 are responsible for chiral symmetry breaking in experiments involving 
 crystallizing a chiral chemical compound from solution .
we showed that secondary nuclei in such stirred crystallization 
 experiments are often easily detached whisker or needle crystallites 
 growing from a mother crystal , and a runaway process involving the 
 formation of secondary nuclei leads to complete chiral symmetry breaking .
water molecules possess a high electric polarizability ; they are 
 electrical dipoles and can be highly affected by the presence of an 
 external field . during dendrite growth , an electric field can produce 
 an ordering of the molecular dipoles and increase the molecular flow 
 toward the dendrite tip .
this increases the growth velocity , decreases 
 the tip radius and disables the generation of side - branches , producing 
 long whiskers .
these effects have long been noted and were studied quantitatively by libbrecht and tanusheva , who measured the tip velocity and found that 
 high voltages could multiply the growth rate more than 10-fold .
we hypothesized that this dendrite growth mechanism should be involved 
 in promoting secondary nucleation .
thus , we undertook laboratory experiments 
 to see whether similarly easily detached forms as in solution crystallization 
 experiments are produced in ice under the influence of an electric 
 field .
we employed a fei quanta 200 environmental scanning electron 
 microscope ( esem ) equipped with a liquid nitrogen cold stage to grow 
 ice in situ at low pressures and at temperatures of 90200 
 k. the microscope was set up so that the cold finger , together with 
 a thermostat , was directly beneath the substrate ( a silicon wafer 
 attached with silver glue ) .
we began by evacuating the chamber in 
 the high - vacuum mode of the microscope ( 6  10 pa ) and lowering the substrate to the working temperature .
we first 
 scanned the uncovered sample substrate , on which we grew an ice film 
 by switching to low - vacuum mode and opening the water input microvalve 
 at a pressure of 40 pa for some seconds .
we found this was the highest 
 pressure at which we could obtain clear images .
we closed the microvalve 
 at or before the point when the substrate temperature increases and 
 can not be maintained at the working temperature , following which we 
 switched back to high - vacuum mode and observed the ice growth in situ . in the electron - microscope chamber ,
a high - voltage electron beam 
 is used for imaging , and as we display in figure 5 , we find that the electric field produces rapid dendrite 
 growth wherever we charge with electrons by imaging .
a typical ice 
 morphology seen under these circumstances is of a form intermediate 
 between whiskers and dendrites , which often takes on the aspect of 
 a palm tree ; see figure 5a . as we zoom out 
 ( figure 5b , c )
, we note that the palm forest 
 is found only where we had been imaging ; outside the area of the electron 
 beam , we find a relatively flat film of ice , while within the imaged 
 zones , three in figure 5c , we find faster ice 
 growth and the ice forest .
growth in an 
 esem : ( a , b ) at t = 170 k , p = 40 
 pa , v = 30 kv , an ice 
the forest displays the morphology intermediate between whiskers 
 and dendrites typically formed with charged ice .
( c ) in an overview , 
 the three zones where we imaged and charged with electrons stand out , 
 showing the increased growth on the background ice film .
we had noted such palmlike forms in previous experiments 
 involving 
 growing ice inside an electron microscope , but had not then realized that the electric field was involved in 
 their production .
these experiments are necessarily qualititative , 
 being performed within the chamber of an unmodified esem , but we find 
 the results suggestive : owing to their geometry , the breakage of these 
 structures on collision is likely and will lead to the formation of 
 new nucleation centers .
such friable morphologies do not only form 
 in ice under electric fields ; snowflakes too have such delicate structures , 
 but an electric field promotes this form of growth .
our minimal physical model of a self - replicating 
 granular material 
 shows how secondary nuclei from such growth can lead to runaway charging
. 
 these effects may be present in ice on earth , in terrestrial thunderstorms , and in astrophysical ices , in the solar nebula , and in thunderstorms on other planets , some of which , 
 for example , on venus , may involve self - replicating granular materials 
 other than water ice . it is conceivable that this dynamics is involved 
 in the formation of the martian geological structures called razorbacks . while ice is clearly 
 the most quotidian example of a friable self - replicating granular 
 material , one that breaks easily and continues to grow , other such 
 materials can be both sought in different astrophysical environments , 
 and also produced in technological contexts .
 we 
 establish that the nonlinear dynamics of collisions between 
 particles favors the charging of an insulating , friable , self - replicating 
 granular material that undergoes nucleation , growth , and fission processes ; 
 we demonstrate with a minimal dynamical model that secondary nucleation 
 produces a positive feedback in an electrification mechanism that 
 leads to runaway charging . we discuss ice as an example of such a 
 self - replicating granular material : we confirm with laboratory experiments 
 in which we grow ice from the vapor phase in situ within an environmental 
 scanning electron microscope that charging causes fast - growing and 
 easily breakable palmlike structures to form , which when broken off 
 may form secondary nuclei . 
 we propose that thunderstorms , both terrestrial 
 and on other planets , and lightning in the solar nebula are instances 
 of such runaway charging arising from this nonlinear dynamics in self - replicating 
 granular matter . 