Basics of thermal conductivity calculations

Transcription

1 Bascs of thermal conductvty calculatons L Ju sometme n Equlbrum Correlaton The bass of calculaton of thermal conductvty and other transport coeffcents usng MD s lnear response theory (Green 1954, Kubo 1957, Mor 1958, Kadanoff and Martn The thermal conductvty ˆκ can be expressed n terms of equlbrum tme autocorrelaton functon of the heat current J q : ˆκ = 1 k B T 2 Ω dτ < J q (J q (τ > (1 Here Ω s the total volume. The nstantaneous heat current s gven by J q = d (E hr (2 dt where E s the mcroscopc energy for each partcle and h s the enthalpy per partcle of the system. If the nteracton s of par potental type, then E = p2 2m V (r j (3 j 1

2 from whch (2 can be wrtten as J q = d (E hr dt = ( (E hv + r [ p F + 1 V j (v v j ] m 2 r j = [(E h 1 (r r j V j ] v 2 r j = (E hv 1 (r r j [ V j (v + v j ] (4 2 r pars On the other hand, when the nteracton s of many-body nature, how to devde the potental energy among the nteractng partcles s not so obvous, and n fact can be qute arbtary. Yet when t s done J q can be derved from the general formalsm: J q = d (E hr dt = ( (E hv + r (v F + E v j r j j = ( (E hv + r ( v E j E + r v j r j r j j = ( (E hv + r j ( E v j r j j (5 (5 provdes an obvous method to calculate the thermal conductvty n a model system, snce all the dynamcal varables n (5 are explctly known n a molecular dynamcs smulaton. Although (1 was derved for a canoncal ensemble, mcrocanoncal ensemble s usually used n MD for t s much easer to mplement, the dfference s only trval snce the phase ponts generated n MD would be n the peak center of densty dstrbuton of the correspondng canoncal ensemble. We wll call ths method equlbrum correlaton method. 2

3 2 Nonequlbrum MD Non-equlbrum molecular dynamcs (NEMD s a formalsm whch bypasses (1, and gve the correct results. It s also based on the lnear response theory, but wth consderable orgnalty and theoretcal nterest n tself, manly, the concept of equvalence of equatons of motons. We know that when a physcal system has Hamltonan H(q, p, the equaton of moton can be readly wrtten down: The Louvlle operator s defned as LA(p, q = so that from Louvlle theorem = q = H/ p (6 p = H/ q ( A q + A p q p ( A H A H q p p q = {A, H} (7 (ṗ, q Tr( (p, q = = Df Dt f Lf (8 t there s f(t = exp(tlf(, where f(t s the densty dstrbuton functonal of phase ponts n Γ space. It can be shown that L s a Hermtan operator. Because the measurement of A(p, q s Ā(t = = Γ Γ dγf(ta(p, q dγ[exp(tlf(]a(p, q (9 3

4 so dā(t dt = = = Γ Γ Γ dγ[l exp(tlf(]a(p, q dγ exp(tlf([ LA(p, q] dγ[exp(tlf(]ȧ(p, q (1 where we have defned the tme dervatve of an arbtary dynamcal varable to be A(p, q = LA(p, q ( A = q + A p q p (11 where the second row conforms both to (7 and the physcal nterpretaton of tme dervatve. From here we can derve the lnear response formulae: suppose the system was orgnally governed by H and s already n thermal equlbrum: f = exp( βh f / t = β exp( βh (L H (12 At t = the perturbaton H was turned on, so H = H + H (13 and L = L + L, f(t = f + f (t (14 The flow equaton (8 remans vald as long as the system s descrbed by a Hamltonan, f t = Lf 4

5 and to frst order n perturbaton strength t can be lnearzed, Multplyng both sde by exp( tl, we have f t = L f + L f (15 [exp( tl f ] t = exp( tl L f whch can be ntergrated to gve f (t = = β = β = β dτ exp((t τl [L f ] dτ exp((t τl f [L H ] dτ exp((t τl f [L H ] dτ exp((t τl f Ḣ (τ (16 For an arbtary dynamcal varable A, the measurement of response to such perturbaton at t > s Ã(t Ā = dγf (ta(p, q Γ = β dτ dγf Ḣ (τ[exp( (t τl A(p, q] = β Ths s the central result of lnear response theory. Γ dτ < A(t τḣ (τ > (17 3 NEMD scheme by Cccott et al As to how (17 can be used to get ˆκ wthout resortng to (1 explctly, let s compare the two equatons: suppose we let A = J q x (p, q n (17, then the rght hand sde of (17 and (1 would look almost smlar, except that one has Ḣ (τ, the other has J q (τ. However, we can 5

6 choose a perturbaton Hamltonan to satsfy Ḣ = λj q x (18 where λ s a very small constant, such that the RHS of (17 and (1 are dentcal except for a constant. Under such a perturbaton, the thermal conductvty ˆκ s smply ( q J ˆκ = lm lm x (t t λ λωt The above formulae assume ˆκ to be an sotropc tensor,.e., ˆκ = κi. When ˆκ sn t sotropc, we can let (19 Ḣ = λj q (t (2 and ( q J j κ j = lm lm (t t λ λωt (21 Whether ˆκ s sotropc or not depend on the local symmetry of the crystal. When the local symmetry s cubc (sold argon or tetrahedral (SC, we can be sure that ˆκ s sotropc from group theory. The problem mmedately follows that how can we choose a H to satsfy (18. However, the answer s obvous once we note that J q s defned by the equaton n (2, so that J q = d (E hr dt H = b (E hr (22 would satsfy (18 automatcally, where b s a small constant. Up to now an deal NEMD scheme had been completely set up. To mplement t we need to put H nto the equaton of moton: r = (p /m (1 + b r p = F (1 + b r ( (E h 1 2 j (r r j V j r b (23 6

7 For sotropc systems, ( b κ = lm Jq (t t b 2 ΩT (24 Ths scheme, however, s only deal and cannot be mplemented on the computer because of the fnte perodcty of the smulaton cell. Imagne the stuaton descrbed by (22 and (23: n order to generate a steady heat current n the system, a constant feld had to be mposed on the whole space,.e., the potental (22 descrbng ths drvng force should be a lnear functon of r (n strct mathematcal sense, ts value rangng from to +. The problem s that on a computer, nstead of a straght lne from to +, we can only generate potental functons of jgjaw shape, wth perodcty of the smulaton cell: dscontnuty would happen on the boundary wth fnte jumps. In other word, there can be no absolute measures of r s that appear n (23. The measures n a peorodc cell can only be of exp(k r, wth k commesurate to the cell, whch means we can only have the fnte k values of the thermal conductvty f we stck to the method. Cccott et al (1978 show that we can study the fnte k thermal conductvty of the system by takng H = λ wth the fnte k heat flux defned by J q k = dentfy A n (17 by J q k k, we have (E hφ, φ = exp(k r (25 ( exp( k r (E hi 1 2 j V j r j p (26 r m J q λβ t k (t k = dτ k < J q k Ω (Jq k (τ > k (27 from where we can get the fnte k autocorrelaton functon of heat flux. And by extrapolatng the fnte k results onto k = we wll arrve at what we want: the macroscopc thermal conductvty. The equaton of moton descrbed by (25 s r = (p /m (1 + λφ (28 ( p = F (1 + λφ λ (29 (E h φ r 1 2 j (φ φ j V j r 7

8 Cccott s scheme s a practcal method to calculate kk. The shortcomng les n the nontrval extrapolaton procedure from fnte k to k =. Another strange feature about (28,29 s that the veloctes r are not smply gven by the momenta p devded by mass. 4 Gllan and Dxon: Equvalence of Equaton of Motons Look at (27 and (28, 29, and let s thnk carefully of where the dffculty les: obvously when we take the lmt k, there must be J q k k n order to satsfy (27. The second term on RHS of (29 ( λ (E h φ 1 r 2 j (φ φ j V j r satsfes ths requrement explctly, snce φ / r, (φ φ j k n k lmt: and we know that the response J q k s proportonal to perturbaton strength appearng n the equaton of moton (n lnear regme. The effect of other two terms, λφ p /m and λφ F, on the other hand, are not so obvous, because there s no drect proportonalty wth k nsde. To study them, let s seperate them out. Consder the followng equaton of moton r = (p /m (1 + λφ p = F (1 + λφ (3 The tme evoluton of an arbtary dynamcal varable A under ths equaton of moton would be where the operators Λ and Λ 1 are Λ Λ 1 DA Dt = (Λ + λλ 1 A (31 = ( p + F m r p = ( p φ + F m r p Note that (3 s not beng descrbed by an Hamltonan, thus although Λ and Λ 1 looks (32 8

9 smlar to the Louvlle operator, they lack the Hermtan property of L. Nevertheless, we can formulate a lnear response theory lke (17 because Hermcty s not requred n the dervaton. Snce flow equaton (8 s no longer vald here, we should use Hesenberg representaton to do the ensemble average for the response to the perturbaton. Take A(t = A (t + A (t, where A (t satsfy DA /Dt = Λ A, the tme evoluton of A n the correspondng unperturbed system. Lnearze (31 nto multply both sde by exp( tλ, there s DA Dt = Λ A + λλ 1 A (t (33 D[exp( tλ A ] Dt = exp( tλ [λλ 1 A (t] whch can be ntergrated to gve So, to lnear order n λ, the measurement Ã(t A (t = λ dτ exp((t τλ [Λ 1 A (τ] Ã(t = Ā(t + λ dτ dγf( exp((t τλ [Λ 1 A (τ] (34 Γ If at t = the system s already n equlbrum, exp((t τλ can be replaced by unty, whch reads Ã(t = Ā(t + λ dτ dγf [Λ 1 A (τ] = Ā(t + λ Γ dτ < Λ 1 A (τ > (35 Plug n the expresson for Λ 1 Ã(t = Ā(t + λ dτ dγf [ Γ = Ā(t λ dτ dγf ( Γ φ ( p m p m φ r A (τ] + F r p A (τ (36 9

10 snce f φ s a constant we should get zero by turnng around both / r and / p, and φ s only a functon of r. Because f p m = k B T f p (36 can be rewrtten as Ã(t = Ā(t + λk B T = Ā(t λk B T dτ dγ Γ dτ dγf Γ f p φ r A (τ φ r A (τ p = Ā(t + λ dτ < Λ 2 A (τ > (37 wth Λ 2 = k B T φ (38 r p Compare (37 wth (35, we see that although perturbaton operators Λ 1 and Λ 2 are algebracally dfferent, the responses they would nduce for an arbtary dynamcal varable are totally dentcal after ensemble averagng. Ths s a truly amazng concluson, snce t proves the exstence of a famly of dynamcal systems wth dfferent equaton of motons, but wth the same macroscopc propertes (n lnear regme. For nstance, Λ 2 can be generated by the followng equaton of moton r p = p /m = F λk B T φ r (39 Although (3 and (39 are dfferent equaton of motons, they are actually equvalent. Now let s go back to thermal conductvty and see how old problems can easly be solved by transferrng the equaton of moton. Snce perturbatons are addtve n lnear regme, (28, 29 are equvalent to r p = p /m ( = F λ (E + k B T h φ 1 r 2 j (φ φ j V j r (4 1

11 As perturbatons terms are all explctly proportonal to k now, we can take the k lmt drectly, and get r = p /m p = F + b ( (E + k B T hi 1 2 wth the k = heat current J q (t gven by (4, and ( b < κ = lm Jq (t > t b 2 ΩT j V j r j r (41 (42 Another way to look at t s to take φ = b r and do equvalence transformaton for (23. All the formalsms above were frst gven by Gllan and Dxon (1983. We know that < J q > = for an equlbrated system. However, Jq (τ and J q (τ are hghly correlated random varables, so nstead of calculatng < J q (t > drectly, we calculate the ensemble average of ther dfference < J q (t J q >, whch must possess much hgher statstcal accuracy as ensemble fluctuatons largely cancel each other. 5 Implementaton on Argon As a model system for the calculaton of thermal conductvty, we choose LJ6-12 argon. There are several purposes n dong ths: To verfy the NEMD algorthm theoretcally derved by Gllan and Dxon. To compare the accuracy between equlbrum correlaton method and NEMD. To see the dfference between lqud state and sold state calculatons. To check the senstvty of results wth respect to cell sze,.e., how many k ponts are enough. In the possble future, put defects nto the system and get emprcal functonal dependence between defect concentraton and thermal conductvty. 11

12 Followng the work by Gllan and Dxon (1983, we mplemented both the equlbrum correlaton and NEMD algorthms on LJ6-12 argon systems. In order to get reasonable results, we must ensure the phase trajectores we generated by MD are hghly smooth, so that the dffernce between the perturbed trajectory and the unperturbed trajectory, whch s very small, won t be nundated by numercal nose. Ths requres us to use a smooth truncated LJ6-12 potental, as any dscontnuty n the potental functon would cause jumps n the phase trajectory: V (r = 4ɛ[(σ/r 12 (σ/r 6 ] r < r γ(r 1 r 2 r < r < r 1 r 1 < r (43 where, for a gven r 1, the constants r and γ are chosen to ensure the contnuty of V and ts frst dervatve at r = r. We ntroduced the cutoff at r 1 = 2σ, same as Gllan and Dxon (1983, and get r = σ, γ = ɛ σ 2 (44 All varables are of double precson n the program to ensure numercal accuracy. Fg.(1 shows a run at T R =.69, DR =.84, NP = 256. The thermal conductvty s n reduced unt, the X axs showng the tme steps, wth one step t = 1ps. A total number of 2745 NEMD runs had been taken wthn the range of 34 runnng steps. For the equlbrum correlaton method, dfferent orgns had been taken (each runnng step after equlbraton as a new orgn, wth correlaton length = 12 steps: the number of effectve orgns s much smaller than the number of as these orgns are hghly correlated, but snce t s a long run we are qute sure that the phase space was fully sampled. The calculated κ for lqud argon at ths condton s.99 ±.3, close to the publshd results by Gllan and Dxon. As we can see from Fg.(1, the comparson between equlbrum correlaton method and the NEMD method by Gllan and Dxon should be consdered as qute satsfactory: not only are the asymptotc value for thermal conductvty close, but also for the whole tme range of the autocorrelaton ntegral. Ths drectly proved the theory developed n prevous sectons of the equvalence transformaton of dynamcal systems, based on whch ths NEMD algorthm 12

13 was put forward. Fg.(2 shows one run at T R =.2983, DR =.9, NP = 256 for sold argon. A total number of 1887 NEMD runs had been taken wthn the range of one mllon steps, the correlaton length beng 45, whch s almost 4 tmes the length needed for the lqud to converge. Snce thermal conductvty n a crystal manly results from the aharmoncty of the lattce,.e., nelastc phonon-phonon collsons, the sze of the smulaton cell s an mportant ssue here, as t drectly controls how many k ponts we have. A run had been taken wth exactly the same condton as Fg.(2, except that NP = 18. The result s shown n Fg.(3: We can see that the equlbrum correlaton curve changed very lttle, but the NEMD curve became shaky after about 15 steps. For overall consderatons we decded that t does not make much dfference between 18 and 256 partcles f we use equlbrum correlaton method. There are several subtle ponts concernng the accuracy of the NEMD method: 1. We must ensure that at t = the perturbaton force be nstantaneously added onto the copy of the orgnal system, not a step later or somethng (for mplementaton n a 5th order Gear method see ball.f. 2. As numercal error (n the fnte-step ntergraton process wll set n for both the perturbed and unperturbed systems, ther trajectores would eventually drft away what they should be. Ths makes the delcate busness of takng the dfferences between two trajectores more and more precarous as tme goes on. We therefore do not recommend NEMD for a very long correlaton run f numercal accuracy can not be guaranteed. The best way to see ths s to test the reversblty of the system after a certan tme, by checkng how much the heat current has changed and compare to the dfference n heat current generated by the perturbaton b. 3. To reduce the error dscussed above, we can ether shorten the ntergraton step, whch s costly, or to ncrease perturbaton strength b to suppress the nose (however, ths wll undermne the valdty of the entre algorthm snce t only works n lnear regme. There would be an optmum value dependng on how many NEMD runs one s gong to take (the more you re gong to take, the smaller the optmum b should be. Usually b s around /σ for argon systems. 13

14 A The Argon Code: ball.f ball.f (65 lnes s a smple molecular dynamcs code for LJ6-12 system wth nteracton potental of Eq.(43 type, able to perform both equlbrum correlaton method and NEMD to get the thermal conductvty of the model system. It uses the temperature couplng method proposed by Berendsen et al (1984, wth couplng tme constant of about 3 ps. Be aware of the fact that temperature couplng must be stopped before NEMD starts, so that the smoothness of the phase trajectory could be guaranteed. ball.f usually generates 5 output fles: temp.out: Ths s the overall record of nstantaneous thermodynamcal varables durng the run, such as the potental energy, pressure and MSD. gr.out: Radal dstrbuton functon. Sometmes you ll fnd B n the fle whch means paragraph breakng, after whch the counter s reset and G(r s re-measured for a new perod of tme. confg: Confguraton fle whch saves the postons and veloctes of partcles, so that later programs can contnue from prevous runs. corr.out: Autocorrelaton functon of heat current n the system, startng from the second row: frst column s the steps, second column s the < J q ( J q (t > functonal, thrd column s the tme ntegral of t. The frst row s the system parameters: temperature, densty and t. nemd.out: Heat current of the perturbed system, startng from the thrd row: ( J x q, J y q, J z q. The frst row s system parameters: temperature, densty and t, the second row s perturbaton strength b : (b x, b y, b z. The nput parameters are put n a fle (suppose called con, whch usually looks lke ths: E 8.. The frst row means 256 partcles wth T R =.2983, DR =.9, the means not readng from the confg fles (f t s 1, then the confg fle of prevous run should be appended behnd 14

15 Thermal conductvty of LJ6 12 system at TR =.69 DR = dashed lne: NEMD algorthm; sold lne: equlbrum correlaton Fgure 1: Flud near trple pont, NP =

16 1.2 Thermal conductvty of LJ6 12 system at TR =.2983 DR = dashed lne: NEMD algorthm; sold lne: equlbrum correlaton Fgure 2: Sold argon, NP =

17 1.2 Thermal conductvty of LJ6 12 system at TR =.2983 DR = dashed lne: NEMD algorthm; sold lne: equlbrum correlaton Fgure 3: Sold argon, NP = 18 17

18 the 4th row. The second row means the total run s 1 steps wth the frst 15 steps as temperature couplng (the couplng constant s 3. ps, and wrtng on temp.out thermodynamcal varables every 5 steps. The thrd row: the frst 1 means equlbrum correlaton turned on, second 1 means NEMD turned on, 45 s correlaton length of both methods (they are the same. The fourth row s perturbaton strength b : (b x, b y, b z. To run, just type: % ball < con & where con s the above condton fle. References [1] H. J. C. Berendsen et al, J. Chem. Phys. 81, 3684 (1984. [2] G. Cccott, G. Jacucc and I. R. McDonald, J. Phys. C 11, (1978. [3] G. Cccott et al, J. Statst. Phys. 21, 1-22 (1979. [4] M. J. Gllan and M. Dxon, J. Phys. C 16, , (1983. [5] M. S. Green, J. Chem. Phys. 22, 398, (1954. [6] L. P. Kadanoff and P. C. Martn, Ann. Phys. (NY 24, (1963. [7] R. Kubo, J. Phys. Soc. Japan 12, 57 (1957. [8] R. Kubo, Rep. Prog. Phys. 29, (1966. [9] H. Mor, Phys. Rev. 112, (