AAthermal Activation in Glassy Fluid
Yuchen Zheng ∗ Department of Physics, Xiamen University,Xiamen 361005, People’s Republic of China (Dated: February 19, 2021) a r X i v : . [ c ond - m a t . s o f t ] F e b bstract In this article, the mechanism of the unexpected high fluidity in SiOx nanowire under modestirradiation was proposed, the high fluidity is attributed to the long lifetime of irradiation-inducedholes, which arise from formation of small polarons. The holes created in irradiation could have along lifetime, and localized in space, such missing of bonding electron could suppress the energybarrier(athermal activation effect) for a Pachner move of the network. The atomic level dynamicsof the system is proposed by interaction of phonon and local configuration, the activation effect wasthen studied with passing rate of corresponding stochastic dynamic equation, calculation shows anexponential dependent of the time-lapse of Pachner move to lifetime of the activation, furthermore,connection between the local configuration time and viscosity of the fluid indicates a strong sen-sitivity of viscosity to lifetime of the athermal activation, such mechanism would give an effectiveinterpretation to the unexpected high fluidity together with the passivation effect of the conductoron the material.
Benefit from the advance in the observation technique in electron microscopy, atomicobservation of material had been achieved[1–6], however, with such high resolution, thebeam-induced phenomena would mix up with the intrinsic nature of the material in in-situ observation, in this way, the beam-induced phenomena have attracted wide researchand discussion[7–9], these beam-induced phenomena can also enlighten invention of newtechnology like nano-fabrication and processing. In recent years, in the investigation ofnano-welding with in-situ electron microscopy(HRTEM) observation, beam-induced atomsmigration has been observed and described[10–13]. In these investigations, the massivefluid-like migration of atoms in the SiOx nanowires under modest irradiation was observed,where the viscosity of such flow under a certain dose rate of electron beam irradiation[13]could be estimated to be 10 P a · s with the assumption of the invariance of the surfacetension, the estimated viscosity was to the viscosity of fused silica which is heated to above2000K, and such high temperature is believed to be un-attainable in HRTEM observation.The fluidity induced by irradiation was further observed to be passivated by pre-depositedmetal nanoparticle[12], in which the nanowire show much less fluidity when pre-depositedwith gold nanoparticle under the same irradiation dose rate.The conventional consideration of beam damage[8, 9] could not offer a detailed interpreta-tion of the observed high fluidity. In the existing discussion, the damage can be divided into2ither irradiation-ion interaction or irradiation-electron interaction, the first is characteristicwith knock-on mechanism[7, 14] where the ion in the material is bombarded by the irradi-ation and exist the material or dislocated in the lattice with momentum transferred by theincoming irradiation. The second is always referred to as the radiolysis, and the sequentialreconstruction of atoms[15]. Accumulation of heat and charge can also be induced in theabove interaction. However, in these considerations, only the long-lived localized damageshave been taken into detailed discussion, while they could be observed either directly inHRTEM or the ordinary spectrum observation, however, localized unstable excitation canalso be created in the irradiation. The effect of such excitation is unable to be analyzed withconventional thermal dynamic methods based on equilibrium, but the following illustrationwould show that there could be intriguing phenomenon which comes out of such excitation.In materials science, as the valent bond electrons form chemical bonds to transfer mo-mentum between ions, and the mass of the electron is about 1/10000 the mass of the ion,when we are dealing with the mechanical response of materials, in often cases we do nothave to come to the level of the electron to deal with the problem, that is, in the solid,the relaxation time of electron is much lower than that of ion, in the region of mechanicalresponse, the interaction of atoms can be dealing with an elastic potential, however, therecould be some extreme cases where the relaxation time of electron is slowed to the scale ofrelaxation time of ions, the elastic potential model under such circumstance would come to afailure, the long-life excitation state could have a great influence on the mechanical responseof materials.Specifically in the SiOx nanowires, in the irradiation, the electron in the material canbe directly knocked or emitted through Auger effect, while in the nanoscale material, theelectron deposited in the irradiation will be less than the electron loss in the irradiation,the net charge accumulated would be positive after irradiation[9]. The positive chargeaccumulated in the irradiation would be holes in the energy band of the material, theseholes created in the irradiation would be localized in lattice and have little contribution tothe conductance through formation of small polaron, the existence of such holes have beenproved through electron spin resonance(ESR)[16], and the lifetime is observed to be extendedto atoms vibrational timescale[17], the localized holes should influence the chemical bondsin the shading area, and the long lifetime of such excitation would enable the influence tomechanical response, the losing of bonding electrons should have a suppressing effect on the3tomic interaction force, to quantify the influence on dynamics of local configuration andthe further mechanical response of the material, the phonon-local configuration interactionshould be introduced.In the silica, the atoms are tightly compact, and in such a situation, the dynamics couldbe established with the phonon as an ingredient. For an atom in the material, the positioncan be expressed with the summarization of amplitude of phonons, by choosing the localcoordinate and only consider in the direction connecting two local equilibriums, in 3 dimen-sion lattice, two T mode and one L mode phonon contribute to the transition, the donationof each mode with different wavelength should be different, nevertheless, the local vibrationmode in high frequency should have less donation for the spacial exponential decay, all theseeffects are included into the response modification function q ( ω i ), and in the neighborhoodof local equilibrium: x ∝ (cid:88) i q ( ω i ) e − ¯ hωikT cos ( θ i ) (1)Where ¯ hω i = k m and θ i is the phase which could be considered to be uniformly distributedin [0 , π ). For each component ω i , the standard derivation is finite, apply the central limittheory, the distribution of x turns to be a Gaussian distribution. P ( x ) ∝ e − mω x kT (2)Where mω is associated with the standard deviation of the distribution of x. Applyingthat ∂θ i ∂t = ω i , the velocity of the atom is:˙ x ∝ (cid:88) i ω i q ( ω i ) e − ¯ hωikT sin ( θ i ) (3)And such distribution is Gaussian too: P ( ˙ x ) ∝ e − m ˙ x kT (4)The average of x and ˙ x will be: (cid:104) x (cid:105) = 0 (5) (cid:104) ˙ x (cid:105) = 0 (6)4nd the covariance of x and ˙ x will be: cov ( x, ˙ x ) = (cid:90) π (cid:89) dθ i π ω j (cid:88) i,j e − ¯ h ( ωi + ωj ) kT q ( ω i ) q ( ω j ) cos ( θ i ) sin ( θ j )= 0 (7)So, the joint probability distribution can be separated, the distribution function of veloc-ity should be a Gaussian distribution independent from the position. By adding up to thepotential which corresponding to the non-harmonic part of the total interactive potentialU(x), we can construct the random dynamic equation: ∂x∂t = − τm ∂U ( x ) ∂x + (cid:115) m B ( t ) (8) (cid:104) B ( t ) B (0) (cid:105) = k B T δ ( t ) (9)Where B ( t ) is a white noise function that fulfills Gaussian distribution, which arises fromthe momentum transfer by the phonon, and τ is a constant which has the unit of s . Thisequation is the famous Smoluchowski differential equation, which describes the dynamics ofa molecule in a potential field with strong damping, such strong damping arises from the factthat the velocity of atoms is always much slower than the speed of sound in the material.The activation effect in the irradiation can be easily introduced to this model by modu-lating the potential with a stepped function. For one dimension problem, the correspondingpotential is simple: U ( x, t ) = U ( x ) f ( t ) (10) f ( t ) = λ ≤ t < t a t a ≤ t < T a (11)Where λ l is the activation coefficient and T a is the average time lapse between twoactivation events. The force field corresponding to the potential is chosen to be piecewiseparabolic: F ( x ) = − k s ( x − x s ) x ≤ − k u ( x − x u ) x ≥ k s > , k u < t ) is:Γ ( t ) = (cid:90) tt dt i Z ( t, t i ) D e − Φ( t,t i ) /D (13)In which: D = τ k B Tm (14)Φ ( t, t i ) r = − x u I u ( ti, t ) + x s I s ( ti, t
0) (15) Z ( t, t i ) = [ Y ( t i ) − x s ] x u π ( I u ( t, t i ) I s ( t i , t )) e u ( ti,t ) (16) Y ( t i , t ) = k s x s I s ( t i , t ) f ( t i ) (17) I s ( t i , t ) = 2 (cid:90) t i t dt (cid:48) e s ( t i ,t (cid:48) ) (18)Λ s ( t i , t ) = − (cid:90) t i t dt (cid:48) k s f ( t (cid:48) ) (19) I u ( t, t f ) = 2 (cid:90) tt f dt (cid:48) e u ( t,t (cid:48) ) (20)Λ u ( t i , t ) = − (cid:90) t i t dt (cid:48) k u f ( t (cid:48) ) (21)Numerical calculation of passing rate Γ ( t ) is shown in Fig. 1, the result is shown inlogarithmic scale, the passing rate changes exponentially shortly after the activation entersor leaves, and gradually saturates to the corresponding Kramers rate. The most importantparameter is the average passage time, which is corresponded to the average time for aPachner move in the network of atoms, is a function of passing rate:¯ t p = T a (cid:82) T a dt Γ ( t ) (22)The integral is calculated numerically and the result is shown in Fig. 2, we can see that ¯ t p show very small plateaus in low activation lifetime and gradually flattening in high activation6 ln[ G (t)] t t a = 1 t a = 4 t a = 1 2 t a = 2 2 FIG. 1. Diagram of passing rate Γ ( t ) in logarithmic scale, the calculation was performed with D = 0 . , k s = − k u = 1 , x u = − x s = 1 , t = − λ = 0 . t a = 1 , , ,
22, the system reachedlocal-equilibrium in t = 0, two Kramers rate is noted with dot line. lifetime, while an exponential decrease lies in between, in this region, the average reachingtime will be strongly affected by a small change of activation timescale resulted from theexponential increasement of Γ ( t ), after Γ ( t ) saturates to Kramers rate, with the increase ofthe activation lifetime, ¯ t p decrease with increase of weight of activation, and such increasewould be smoother than the exponential decrease. Furthermore, the divergence between thetwo Kramers rate would increase with decrease of temperature, that is, the system wouldbehave more abnormal in lower temperature.The connection between a timescale variable describes the atomic movement with theviscosity of the material is generally discussed[19, 20], the definition of viscosity is associatedwith a Maxwell relaxation time τ M : τ M = ηG ∞ (23)Where G ∞ is the infinite frequency shear module and in strong glass like fused silicaglass in this case, τ M ∝ τ LC , the τ LC is the time of local configuration, and τ LC ∝ ¯ t p , whilethe divergence of τ M with τ LC only happen in fragile glass, the viscosity of the activatedmaterial should be: η a = ¯ t p t η (24)7 lntp t a lntp t a FIG. 2. Diagram of average passage time ¯ t p with logarithmic scale, T a = 1000 and other param-eters are the same as Fig.1, a small plateau shows in the first stage, the curve further saturatedwith increase of t a , the plateau in the early stage is shown in top right. In which t is the corresponding pristine average passing time in the non-activated case.Such relation indicated that change of viscosity by activation can be easily observed inrheology experiment.In the experimental scenario, the non-exponential dependence of time-lapse of activationT indicates the activation effect cannot be exponentially suppressed by the density of holesinstead of the lifetime of holes, which means the dose of irradiation would not serve to bea threshold in the changing of viscosity. And the lifetime of holes created by irradiationcould be suppressed by the mutual repulse interaction, electric field induced by accumula-tion of charge, interaction with phonon, and nearby free-electron donator, this explains thepassivation effect of pre-deposited conductor nanoparticle observed in the experiment.The low viscosity observed in the irradiated SiOx nanowire cannot be simulated throughthe conventional ab-initio way, because the most practical method in the simulation wasbased on the Born-Oppenheimer approximation, the exact trajectory of carrier wave functionalong the time axis is neglected, and in the most cases, such approximation works well. Butin some cases, like solving the movement of light atoms like hydrogen in the water molecule,such approximation show observable diversion with the fact[21], furthermore, this paperproposes a new scenario where the approximation breaks, the movement of electron couldbe localized and slowed to scale of the movement of atoms, the dynamics of atoms could bestrongly influenced by the lifetime of the electron.8n the article, the beam-induced fluid-like migration of atoms is interpreted with ather-mal activation effect in local configuration-phonon interaction, further calculation shows anexponential dependence of the local configuration lifetime with activation lifetime, this offersa good interpretation to the passivation effect of the pre-deposited metal nanoparticle. 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