Kinetic Energy-Based Temperature Computation in Non-Equilibrium Molecular Dynamics Simulation

Transcription

1 Copyrght 202 Amercan Scentfc Publshers All rghts reserved Prnted n the Unted States of Amerca Journal of Computatonal and Theoretcal Nanoscence Vol. 9, , 202 Knetc Energy-Based Temperature Computaton n Non-Equlbrum Molecular Dynamcs Smulaton Ran Xu, Bn Lu 2, Xaoqao He 3, and Dechang L AML, Department of Engneerng Mechancs, Tsnghua Unversty, Bejng 00084, Chna 2 Center for Nano and Mcro Mechancs, Tsnghua Unversty, Bejng 00084, Chna 3 Department of Buldng and Constructon, Cty Unversty of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong, Chna A knetc energy-based approach s developed to obtan the correct local temperature of nonequlbrum systems. We have demonstrated that as a temperature measure, the average dsturbance knetc energy of a sample s more applcable than ts average total energy, and the latter has been wdely used n most molecular dynamcs software. However, t s proved and demonstrated by our smulaton example that the average dsturbance knetc energy s sample-sze dependent. By usng a smple equlbrum system as a thermometer, we propose a calbraton approach to elmnate ths sample-sze effect. A vbratng atomc bar s used as an example to test the valdty and convergence of varous temperature defntons. Keywords: Temperature, Non-Equlbrum, Molecular Dynamcs.. INTRODUCTION Temperature s one of the fundamental physcs concepts ndependent of materals. It s used to quanttatvely measure the extent of hotness or coldness, and represents the ntensty of the thermal moton of molecules n mcroscopc theory. Recently, researchers have found that how to compute the temperature, or the defnton of temperature, strongly nfluences molecular dynamcs (MD smulaton, n whch the veloctes of atoms are contnuously adjusted accordng to varous temperature-control algorthms. For example, n modelng mult-walled carbon nanotubes as ggahertz oscllators, dfferent oscllaton propertes, ncludng qualty factor Q, were obtaned at the same reference temperature (e.g., Refs. [ 3], whch makes smulaton results controversal. In equlbrum statstcal mechancs, the absolute temperature s proportonal to the average knetc energy. Rugh 4 5 developed a method to determne temperature n Hamltonan dynamcal systems, whch has been used and promoted by many researchers (e.g., Refs. [6 8]. Nevertheless, t has not yet been ascertaned whether all components of knetc energy contrbute to temperature. Many researchers (e.g., Ref. [9] omt the rgd translatonal knetc energy from temperature calculaton, whle some (e.g., Ref. [0] smply leave out translaton of the mass center durng smulaton. However, there are stll some unanswered questons n computng knetc energy-based temperature, whch Author to whom correspondence should be addressed. are addressed n ths paper: Should rgd rotatonal and mechancal vbraton knetc energy also be excluded n temperature calculaton? How to extract knetc energy related to pure thermal moton? The answers to these questons are crucal for non-equlbrum MD smulatons snce many MD smulatons am to model dynamc processes and there s an urgent need to establsh a defnton of temperature for non-equlbrum states. Although many prevous studes provde varous defntons of temperature out of equlbrum (e.g., Refs. [8, 5], there s no consensus as yet. In ths paper, we propose a smple temperature computaton method for non-equlbrum MD smulatons, whch s knetc energy-based and s easy to mplement n MD. In order to nvestgate applcabltes of varous temperature defntons, we frst clarfy four necessary condtons for a correct temperature defnton: Condton : If a system s n thermodynamc equlbrum, then the defnton should yeld the same value at dfferent sample postons. Condton 2: If a system s n thermodynamc equlbrum, the computed temperature should be almost ndependent of sample sze when ths sze s suffcently large. Condton 3: The temperature based on ths defnton should be ndependent of the choce of reference frame. Condton 4: The temperature defnton should be reducble to the classcal defnton n the smple equlbrum stuaton. The paper s organzed as follows. Secton 2 and Secton 3 dscuss the necessty and measures of flterng J. Comput. Theor. Nanosc. 202, Vol. 9, No /202/9/00/006 do:0.66/jctn

2 Knetc Energy-Based Temperature Computaton n Non-Equlbrum Molecular Dynamcs Smulaton Xu et al. out the knetc energy of the rgd-body moton and mechancal vbraton, respectvely, n temperature computaton. The conclusons are summarzed n Secton WHY AND HOW TO FILTER OUT THE KINETIC ENERGY OF THE RIGID-BODY MOTION IN TEMPERATURE COMPUTATION For an N -atom system, the Hamltonan can be expressed wth the square of momenta p, and then the wdely used temperature defnton s T = 2 3k B N N p 2 = 2 H ( 2m 3k B H Ds+R, and H Ds are averages of the total knetc energy, knetc energy wthout the rgd body translaton part, and dsturbance knetc energy excludng both the rgd body translaton and rotaton parts of the sample, respectvely. In the followng, we use a rotatng atomc ball wth 8630 atoms, as schematcally shown n Fgure (a, to test ther applcabltes to serve as a temperature measure. The nteratomc potental s the Lennard Jones potental V r = 4 /r 2 /r 6, where s the depth of the potental (a ω where p denotes the temporal average, k B s the Boltzmann constant, m s the mass of atom, and H s the average knetc energy for each atom. To determne the temperature of a gven pont n a system, we need to compute the average knetc energy of ts local neghborng sample. A sample contans l atoms, l N, and ther velocty can be decomposed nto three parts as v = v C + ˆr + v Ds (2 where ˆr s the atom poston vector relatve to the mass center of the sample, v C s the translatonal velocty of the sample mass center, s the average rotatonal velocty around the sample mass center, v Ds s the dsturbance velocty. These can be computed as v C = ˆr m v = l m m v (3 m ˆr 2 I ˆrˆr (4 Correspondngly, there are three possble types of average knetc energy: and H Total = l = 2l ( 2 m v v H Ds+R = 2l m v Ds v Ds + J C + Mv C v C ( H Ds = 2l m v Ds v Ds + J C (5 (6 m v Ds v Ds (7 where M s the total mass of the sample, and J C = l m ˆr 2 I ˆrˆr s the rotatonal nerta. Obvously, H Total, (b. Normalzed Knetc Energy H (c Normalzed Knetc Energy H ω Dsturbance Knetc Energy H Ds d Dsturbance+Rotaton Knetc Energy H Ds+R Total Knetc Energy H Total Sample Center Poston d/σ Dsturbance Knetc Energy H Ds Dsturbance+Rotaton Knetc Energy H Ds+R Atom Number N Sample Fg.. (a A snapshot of the solated rotatng atomc ball n thermodynamc equlbrum. The normalzed average knetc energy as a functon of the sample poston (b and the sample sze (c. ω 429 J. Comput. Theor. Nanosc. 9, , 202

3 Xu et al. Knetc Energy-Based Temperature Computaton n Non-Equlbrum Molecular Dynamcs Smulaton well, s the dstance at whch the nteratomc potental s zero, and r s the dstance between the atoms. The ball s smulated as an solated system. Intally, each atom has a random thermal velocty and the system has a rotatonal velocty. After a suffcent amount of tme has passed, ths solated system fnally reaches a state of thermodynamc equlbrum, n whch all physcal state varables reman statstcally unchanged over tme but the rotatonal momentum s non-zero and conserved. For ths system n thermodynamc equlbrum, the temperature should be unform, as stated earler n Condton. Normalzed by the value of the whole system, the three types of average knetc energy from dfferent sample postons of the rotatng ball are shown n Fgure (a; each sample conssts of 860 neghbor atoms. It s found that the average total knetc energy of samples H Total has sgnfcantly dfferent values for dfferent sample postons, whch ndcates that the average total knetc energy should not be used as the measurement of temperature n some cases. Then we study the effect of sample sze on values of average knetc energes H Ds and H Ds+R. As shown n Fgure (b, dfferent-szed concentrc sphere samples are tested. The results ndcate that wth ncrease n sample sze, H Ds+R does not converge but t contnuously ncreases. It s obvous that as temperature measures, H Total and H Ds+R volate Condton 3, because dfferent reference frames lead to dfferent H Total and H Ds+R. In contrast, average dsturbance knetc energy H Ds seems a better choce to measure the temperature. However, H Ds s only the average knetc energy excludng the rgd moton. It mght contan the mechancal vbraton knetc energy, and predct a wrong temperature. 3. WHY AND HOW TO FILTER OUT KINETIC ENERGY OF MECHANICAL VIBRATION IN TEMPERATURE COMPUTATION In general stuatons, knetc energy of a sample ncludes three parts: the rgd moton part, the mechancal vbraton part, and the thermal moton part. It s easy to remove the rgd moton part from knetc energy computaton by choosng a comovng and corotatng frame. However, t s not straghtforward to decouple the mechancal vbraton part and the thermal moton part. We note the fact that when the sample becomes smaller, the fracton of the mechancal vbraton part n the average dsturbance knetc energy H Ds becomes lower untl t can be gnored. Therefore, when H Ds s used to measure the temperature, a smaller sample s preferred to flter out the contrbuton from mechancal vbraton. On the other hand, f the temperature vares spatally n non-equlbrum systems, small samples should also be used to obtan the local temperature of a gven pont. 3.. Sze Dependence of the Average Dsturbance Knetc Energy of Small Samples To demonstrate the sample-sze dependence, we frst dscuss an extreme case, where the sample has only one atom n t. The velocty of ths sample equals to the velocty of the atom,.e., v C = v. By removng the rgd moton, the average dsturbance knetc energy H Ds becomes zero, whle H Ds s obvously non-zero for any larger samples. More generally, we may also prove that the smaller sample corresponds to lower average dsturbance knetc energy. Assume there s a large sample wth atoms. We may use t to compute ts average velocty v, or dvde ths large sample nto two small samples: Sample wth mass M and Sample 2 wth mass M 2, and compute the correspondng average veloctes v and v 2, respectvely. The average dsturbance knetc energes n these two ways of computaton are H Ds large sample = = = 2 m v v 2 ( 2 m v M +M 2 2 m v 2 M +M 2 2 m v 2 ( 2 m v and H Ds small sample = H sample Ds Ds +Hsample2 2 = [ N 2 N 2 m v 2 ( N 2 ] NM 2 m v + [ 2 N =N + 2 m v 2 ( 2 ] NM 2 =N + 2 m v = 2 m v 2 ( N 2 m M 2 v M 2 2 ( (8 2 m v (9 =N + The dfference between Eqs. (8 and (9 s H Ds Ds smallsample Hlargesample = 2 P M 2 P 2 M 2 M M 2 M +M 2 0 (0 where P = N m v and P 2 = =N + m v are momentums of Sample and Sample 2, respectvely. It should be ponted out that the above dervaton also J. Comput. Theor. Nanosc. 9, ,

4 Knetc Energy-Based Temperature Computaton n Non-Equlbrum Molecular Dynamcs Smulaton Xu et al. holds for other parttons wth many smaller average volume elements. Equaton (8 mples that the smaller local average volume has lower local dsturbance knetc energy, unless P M 2 P 2 M = 0. To further demonstrate the sample-sze dependence, a system n smple equlbrum state wthout any rgd moton (Fg. 2(a s nvestgated. Fgure 2(b ndcates that for the smaller sample, H Ds s dependent on the sample sze N sample,.e., the number of atoms n a sample. As we know, for ths smple equlbrum case, the classcal defnton of temperature, T = 2/3k B H Total, s wdely accepted (of course, ths defnton may fal for other complex stuatons, as dscussed earler. It s also found that although H Ds for suffcently large sample converges to H Total, H Ds of the smaller sample s sgnfcantly lower than H Total. Therefore, drectly usng H Ds leads to a lower temperature estmaton and Condton 4 s volated A Calbraton Approach to Avod Sample-Sze Dependence n Local Temperature Computaton for Dynamc Problems We propose a calbraton approach to compute temperature correctly n molecular dynamcs smulatons. The essental (b Normalzed Knetc Energy H/ε (a H Ds H Ds H Ds No rgd moton T 3 =2.496 ε/k β T 2 =2.29 ε/k β T =2.072 ε/k β Atom Number N sample H Total H Total H Total Fg. 2. (a Schematc dagram of a system n smple equlbrum wthout rgd moton; (b The average dsturbance knetc energy, H Ds,asa functon of sample sze. dea s based on the assumpton that temperatures of two samples are the same f ther sample sze (N sample and average dsturbance knetc energy (H Ds are dentcal. Consderng that a smple equlbrum system wthout rgd moton and mechancal vbraton has a wdely accepted temperature defnton, t can be used as a thermometer or benchmark to calbrate temperature n non-equlbrum states. Fgure 2(b s a temperature contour map obtaned from a system n equlbrum wthout rgd moton, and can serve as a calbraton chart to predct the temperature for the non-equlbrum state. If average dsturbance knetc energy H Ds (N sample and sample sze N sample are known, then the rght temperature can be obtaned by nterpolaton (Fg. 2(b. Wth ths calbraton treatment, Condton 4 s satsfed. Therefore, the defnton of temperature based on the calbrated average dsturbance knetc energy has wde applcablty, as the followng dynamc example demonstrates A Dynamcal Example Under Thermostat We smulate the dynamc behavor of an atomc bar n sothermal envronment (Fg. 3(a whch ncludes 8000 atoms wth a harmonc nteratomc potental /2 k r j b 2, where k s the harmonc force constant and b s the equlbrum bond length. The bar s ntally under unaxal statc tenson, and s n thermomechancal equlbrum at T = 0 5 kb 2 /k. The loadng s then suddenly released, and the bar starts to vbrate, as shown n Fgure 3(a. It s expected that ths system wll experence macroscopc quas-harmonc vbraton and related physcal quanttes wll exhbt a regular perodc curve wth respect to tme, at least n the frst several cycles. Fgures 3(b d show the knetc energy of the whole system as a functon of the normalzed tme, and four energy/temperature control modes wth dfferent number of samples are adopted to test the valdty and convergence of varous MD schemes. If ths atomc bar s smulated as an solated system n MD smulaton,.e., the total energy (ncludng both knetc and potental parts s conserved, then the correspondng curve s really a regular perodc curve as denoted by the black sold lne. Accordng to our common sense, vbraton behavor of the solated system durng the frst several cycles s smlar to that of the system under thermostat, whch provdes a reference n nvestgatng varous temperature controlled schemes. Because there are mechancal vbraton and thermal dsturbance moton n ths system, smaller samples are preferred to accurately obtan and control the temperature. On the other hand, too small a sample mght lose ts statstcal feature. A good smulaton should be ndependent of the arbtrary choce of researchers, such as the number of samples when evaluatng and controllng temperature. The bar s then dvded nto dfferent number of samples, such as, 5, 5, 40 and 80, to study the convergence. The Nose 6 7 Hoover temperature control method developed by Nose 43 J. Comput. Theor. Nanosc. 9, , 202

5 Xu et al. Knetc Energy-Based Temperature Computaton n Non-Equlbrum Molecular Dynamcs Smulaton (a F y x F (c Normalzed Knetc Energy H/kb Free TC on 5 sample sets 2.4 TC on 5 sample sets 2.0 TC on 40 sample sets TC on 80 sample sets Normalzed Tme t k m (b 2.4 Normalzed Knetc Energy H/kb Free TC on sample set TC on 5 sample sets TC on 5 sample sets TC on 40 sample sets TC on 80 sample sets Normalzed Tme t k m (d 2.8 Normalzed Knetc Energy H/kb Free TC on 5 sample sets TC on 5 sample sets TC on 40 sample sets TC on 80 sample sets Normalzed Tme t Fg. 3. (a Schematc dagram of a vbratng atomc bar; The normalzed knetc energy of a bar versus the normalzed tme wth dfferent numbers of Nose Hoover thermostat controllng (TC samples based on average total knetc energy (b, average dsturbance knetc energy (c and the calbrated average dsturbance knetc energy (d. and Hoover, 8 9 s used to smulate an sothermal envronment by adjustng the correspondng average knetc energy n each sample. If the average total knetc energy H Total s adopted n the thermostat, as n many wdely used MD softwares, curves for dfferent numbers of samples shown n Fgure 3(b devate sgnfcantly from the curve for the solated system, whch seems unreasonable. If we drectly use the average dsturbance knetc H Ds n temperature control, t can be observed from Fgure 3(c that the smulaton s not converged wth respect to the sample sze. The correspondng curves exhbt many local zgzags and ther ampltudes ncrease wth reducton of sample sze, whch makes the curves dfferent from the reference curve for solated system. The reason s that drectly usng H Ds always underestmates the local temperature and the thermostat tends to nput the energy at ntervals to keep the desred temperature. In contrast, f the calbrated average dsturbance knetc energy and small samples are adopted n the thermostat, as shown n Fgure 3(d, MD smulatons yeld smooth regular perodc curves, whch converge to the reference curve of solated system. Therefore, our suggested k m calbraton for the H Ds -based defnton of temperature s sutable for more complex stuatons n MD smulatons. 4. CONCLUSIONS To correctly obtan the local temperature n nonequlbrum systems, we propose a temperature defnton based on the calbrated average knetc energy of a sample. The valdty of ths and other knetc-energy-based temperature defntons are dscussed n ths paper. The followng related ssues have been studed and clarfed. ( The knetc energy of a sample can frst be dvded nto the rgd moton part and the dsturbance part. However, t s not straghtforward to further decouple the latter nto mechancal vbraton part and thermal moton part. (2 The temperature should only be related to the knetc energy of thermal moton; therefore, n a dynamc system, the smaller sample s preferred to flter out the contrbuton from mechancal vbraton. However, t s proved and ndcated by our smulaton example that the average dsturbance knetc energy s sample-sze dependent. J. Comput. Theor. Nanosc. 9, ,

6 Knetc Energy-Based Temperature Computaton n Non-Equlbrum Molecular Dynamcs Smulaton Xu et al. (3 Ths unwanted sample-sze effect n temperature computaton can be elmnated by calbratng wth a correspondng smple equlbrum system, and usng the assumpton that temperatures of the two samples are the same f ther sample sze and average dsturbance knetc energy are dentcal. Ths approach has been valdated n the dynamc example of ths paper. Acknowledgments: The authors acknowledge the support from the Natonal Natural Scence Foundaton of Chna (grant nos , , , and and Natonal Basc Research Program of Chna (973 Program, grants nos. 2007CB and 200CB References. Q. S. Zheng and Q. Jang, Phys. Rev. Lett. 88, 3 ( Q. S. Zheng, J. Z. Lu, and Q. Jang, Phys. Rev. B 65, 6 ( C. C. Ma, et al., Nanotechnology 6, 253 ( H. H. Rugh, Phys. Rev. Lett. 78, 772 ( H. H. Rugh, J. Phys. A-Math. Gen. 3, 776 ( G. P. Morrss and L. Rondon, Phys. Rev. E 59, R5 ( J. Casas-Vazquez and D. Jou, Rep. Prog. Phys. 66, 937 ( J. G. Powles, G. Rckayzen, and D. M. Heyes, Mol. Phys. 03, 36 ( M. J. Ulne, D. W. Sderus, and D. S. Cort, J. Chem. Phys. 28, 7 ( D. Van der Spoel, et al., J. Comput. Chem. 26, 70 ( E. Bertn, et al., Phys. Rev. E 75, 6 ( L. Berther and J. L. Barrat, J. Chem. Phys. 6, 6228 ( A. V. Popov and R. Hernandez, J. Chem. Phys. 26, 6 ( J. L. Garden, J. Rchard, and H. Gullou, J. Chem. Phys. 29, 0 ( W. G. Hoover and C. G. Hoover, Phys. Rev. E 77, 8 ( S. Nose, J. Chem. Phys. 8, 5 ( S. Nose, Mol. Phys. 52, 255 ( W. G. Hoover, Phys. Rev. A 3, 695 ( W. G. Hoover, Phys. Rev. A 34, 2499 (986. Receved: 23 March 20. Accepted: 7 May J. Comput. Theor. Nanosc. 9, , 202