D. Keffer ME 64 Det. of Materals cence & Engneerng Unversty of ennessee Knoxvlle Evaluatng hermodynamc Proertes n LAMMP Davd Keffer Deartment of Materals cence & Engneerng Unversty of ennessee Knoxvlle date begun: March 06 date last udated: March 4 06 able of ontents I. Purose of Document... II. ecfyng the hermodynamc tate... III. Evaluatng hermodynamc Proertes... III.A. Instantaneous Proertes... number of artcles of comonent... mole or atom fracton of comonent x... volume of the system... temerature... ressure... 4 knetc energy KE... 5 otental energy PE... 5 nternal energy U (or total energy E)... 6 enthaly H... 6 III.B. Fluctuaton Proertes (heat caacty sothermal comressblty etc.)... 6 III.. Proertes ncludng Entroy... 8 entroy... 8 free energes A and G... 9 chemcal otental of comonent µ... 9 entroy dfferences... 9
D. Keffer ME 64 Det. of Materals cence & Engneerng Unversty of ennessee Knoxvlle I. Purose of Document he urose of ths document s to rovde a ractcal ntroducton to the evaluaton of thermodynamc roertes n LAMMP. he notes begn wth some formal theory and conclude wth ractcal mlementaton. From an undergraduate course n thermodynamcs we have a lst of thermodynamc roertes of nterest. Here we revew them n three categores: () nstantaneous roertes mmedately avalable n LAMMP () roertes that can be relably obtaned wth a lttle addtonal work and () roertes nvolvng entroy. II. ecfyng the hermodynamc tate We begn by recallng the Phase Rule that states that the number of thermodynamc degrees of freedom DOF of a system are related to the number of hases (φ) and the number of comonents. DOF φ + () For a sngle comonent sngle hase system DOF meanng that the thermodynamc state s fully secfed by two varables for examle temerature and ressure or and the number densty ρ. In fact any two varables can be used to secfy the state. For examle a state could be secfed by rovdng the enthaly H and the ρ but that s just not very ractcal. he Phase Rule s based on ntensve roertes thus t does not consder the secfcaton of the number of atoms or molecules as one of the state varables. In the smulaton of a sngle comonent sngle hase system we therefore secfy three varables n order to fully secfy the system. Below we lst the four routnely accessble ensembles based on the choce of varables. In the common notaton smulaton volume an extensve varable s secfed but ths does not trouble us snce the number densty s related to the system volume va ρ. E mcrocanoncal ensemble canoncal ensemble sobarc-sothermal ensemble H sobarc-senthalc ensemble In undergraduate thermodynamc courses we ordnarly use the symbol U to reresent the nternal energy whereas n statstcal mechancs the symbol E s used. Here U E. In a multcomonent sngle-hase system the hase rule rovdes that each addtonal seces adds an addtonal degrees of thermodynamc freedom DOF +. ycally ths means that we secfy the number of each tye of atom or molecule whch secfes not but only - degrees of freedom. It s easest to see ths n terms of the ntensve resentaton of the of each seces namely the mole fracton (or equvalently atom fracton) of comonent x. x j j ()
D. Keffer ME 64 Det. of Materals cence & Engneerng Unversty of ennessee Knoxvlle nce the mole fractons are constraned to sum to unty x j j () only - of the mole fractons are ndeendent. hus there reman two addtonal thermodynamc degrees of freedom such as and or and ρ. As you ncrease the number of hases the thermodynamc degrees of freedom decrease. In a sngle comonent two-hase system DOF. ecfyng one varable say the temerature fully defnes the thermodynamc state. In terms of vaor-lqud equlbrum secfyng the bolng temerature secfes the vaor ressure whch s the ressure at whch the vaor and lqud can co-exst at that temerature. III. Evaluatng hermodynamc Proertes III.A. Instantaneous Proertes he soluton of the classcal equatons of moton results exclusvely n a set of trajectores whch are comrsed of atomc ostons r and veloctes v (or equvalently momenta ) as a functon of tme t. Every thermodynamc roerty s formulated as a functon of these trajectores. nce classcal thermodynamcs s a contnuum theory formulated wthout knowledge of the atomc nature of matter t does not rovde relatonshs between thermodynamc quanttes and atomc trajectores. For that we requre statstcal mechancs sometmes called statstcal thermodynamcs when lmted to equlbrum systems. We make no attemt here to derve any of these relatonshs. We smly resent some relatonshs and lnk them to varables avalable n LAMMP. number of artcles of comonent In the four ensembles descrbed above the number of artcles (atoms or molecules) s held constant. herefore ths extensve varable s always an nut to the smulaton. It s a constant and has no fluctuaton durng the course of the smulaton. mole or atom fracton of comonent x nce the number of artcles s a fxed nut er equaton () the mole fractons are also fxed nuts. volume of the system In the E and ensembles the volume of the smulaton cell s a fxed nut. It does not fluctuate wth tme. In the and H ensembles the volume fluctuates n resonse to the barostat. temerature he temerature s related to the knetc energy KE va the equartton theorem. he knetc energy s related to the veloctes va classcal mechancs.
D. Keffer ME 64 Det. of Materals cence & Engneerng Unversty of ennessee Knoxvlle k KE m v (4) Unlke the volume the temerature fluctuates n all four ensembles. he and ensembles n whch the temerature s secfed are controlled by thermostats that do not kee the temerature exactly at the set ont temerature at all tmes. Instead a roer thermostat allows the temerature to fluctuate about the set ont temerature n a manner that s stll caable of rgorously generatng trajectores that corresond to the ensemble. ressure In three-dmensons the ressure tensor s a symmetrc x tensor wth each comonent defned by a knetc and otental contrbuton KE + (5) β β PE β he knetc contrbuton to the β element of the ressure s KE β mv v β (6) nce the veloctes n dfferent dmensons are not correlated when β the average value of the s zero. However β we have twce the knetc energy n that dmenson. KE β δ β KE β mv (7) If use the equartton theorem relate the knetc energy n terms of the temerature and average over all three dmensons we have the ressure from the deal gas law where the otental energy s zero k KE m v for deal gas only (8) hus the otental energy contrbuton to the ressure can be seen as a devaton from the deal gas. PE β r f β (9) 4
D. Keffer ME 64 Det. of Materals cence & Engneerng Unversty of ennessee Knoxvlle When β the average value of the PE β s not necessarly zero deendng on the state of stress wthn the system. If the ressure tensor s not of nterest the scalar ressure reorted by LAMMP s the thermodynamc ressure r f (0) mv + Lke the temerature the ressure fluctuates n all four ensembles. he and H ensembles n whch the ressure s secfed are controlled by barostats that do not kee the ressure exactly at the set ont temerature at all tmes. Instead a roer barostat allows the ressure to fluctuate about the set ont ressure n a manner that s stll caable of rgorously generatng trajectores that corresond to the ensemble. knetc energy KE he knetc energy s not a tradtonal thermodynamc varable. However t s an mortant roerty n the atomc-level descrton of matter. he KE s defned by the sum of the knetc energy of all artcles n the system as gven n equaton (4). In ths work the knetc energy s defned exclusvely as a functon of veloctes. Exlctly the knetc energy s not a functon of artcle ostons. If the system does not have net zero momentum then there s a center of mass velocty that exsts n the smulaton. he knetc energy can then be broken nto two arts. Knetc energy of the center of mass of the system and the ecular knetc energy whch s the sum of knetc energy of each artcle based on veloctes relatve to the system center of mass velocty. Imortantly when the equartton theorem s used to calculate temerature t reles on ecular veloctes. ee the notes on thermostats to roerly account for ths effect. otental energy PE he otental energy s not a tradtonal thermodynamc varable. However t s an mortant roerty n the atomc-level descrton of matter. he PE s defned by the sum of all nteractons between all artcles n the system as gven by the nteracton otental. he Lennard-Jones otental Embedded-Atom Method otental and OPL-aa otental are all examles of nteracton otentals that defne the otental energy. In ths work the otental energy s defned exclusvely as a functon of ostons. Exlctly the otental energy s not a functon of artcle veloctes. 5
D. Keffer ME 64 Det. of Materals cence & Engneerng Unversty of ennessee Knoxvlle nternal energy U (or total energy E) he nternal energy s the sum of the knetc and otental energes. It s also called the total energy. Often n thermodynamcs we here of the excess nternal energy whch subtracts the KE form the total energy. However n general the nternal energy ncludes both contrbutons. U E KE + PE () In the E ensemble the nternal energy should be constant as t s a measure of the conservaton of energy of the classcal equatons of moton. It truth t fluctuates based on the numercal aroxmatons to the soluton of these ordnary dfferental equatons. In the other three ensembles the nternal energy fluctuates. enthaly H he enthaly s defned as the sum of the nternal energy and the roduct of ressure and volume. H U + () All of the quanttes lsted n ths secton can evaluated at each nstant n tme yeldng so called nstantaneous roertes. Of course what s tycally mortant s to reort the average roertes. he average can be comuted n a ost-rocessng mode usng the nstantaneous values rnted out n the LAMMP outut fle. Alternatvely the averages can be comuted n the LAMMP nut fle usng the fx ave/tme command. ee for examle the block averagng examles on the course webste that comute both the average as well as the standard error usng the fx ave/tme command. III.B. Fluctuaton Proertes (heat caacty sothermal comressblty etc.) here are a varety of thermodynamc roertes that are based on the artal dervatve of roerty X wth resect to roerty Y holdng roerty Z constant. ome common thermodynamc roertes of ths form nclude the constant-ressure heat caacty sothermal comressblty β and the thermal exansvty. the H (.a) β (.b) (.c) 6
D. Keffer ME 64 Det. of Materals cence & Engneerng Unversty of ennessee Knoxvlle he exresson gven above are thermodynamc defntons. tatstcal mechancs rovdes an elegant theoretcal way to evaluate these roertes based on the fluctuatons of roertes. hese fluctuatons are based on the varance or covarance of the nstantaneous thermodynamc roertes dscussed n the revous secton. For examle n the ensemble the constant ressure heat caacty s roortonal to the varance of the enthaly < σ > H (4.a) k he sothermal comressblty s obtaned through the varance of volume β < σ > (4.b) k he thermal exanson coeffcent s based on the covarance of volume and enthaly of the < σ H k ( )( ) > (4.c) he angled brackets ndcate an ensemble average namely an average over both all artcles and tme. ote that ths varance s based on nstantaneous values corresondng to the square of standard devaton of the varables. Exlctly these fluctuatons do not corresond to the square of the standard error generated by the block averagng rocedure. hese fluctuatons seem to rovde an effcent method to obtan these thermodynamc roertes snce we can obtan all these three roertes from one sngle MD smulaton at the desred state ont. he formulae gven n equaton (4) are for the ensemble. Other formulae exst for other ensembles. We could also calculate β and usng a second rocedure namely centered fnte dfference usng average values of roertes over multle smulatons. In ths aroach one runs three smulatons for each thermodynamc roerty. For examle for the heat caacty one runs a smulaton at the state ont defned by () then two other smulatons at (+ε) and (-ε). he choce of ε s made so that one can observe statstcally vald dfferences n the enthaly between the state onts. hus the heat caacty can be determned as ( + ε ) H ( ε ) H (5.a) ε mlarly the sothermal comressblty and thermal exansvty can be exressed as ( ) ( + ε ) ( ε ) β (5.b) ε 7
D. Keffer ME 64 Det. of Materals cence & Engneerng Unversty of ennessee Knoxvlle ( ) ( + ε ) ( ε ) ε (5.c) A detaled comarson of these aroaches s gven n the followng reference. Wang Q. Keffer D.J. Petrovan. homas J.B. Molecular Dynamcs mulaton of Polyethylene erehthalate Olgomers J. Phys. hem. B. 4() 00. 786 795 do: 0.0/j90976j. her analyss found that the fnte dfference aroach was markedly more accurate. Any advantage thought to be ganed by effcency of obtanng the result from a sngle smulaton was lost snce n order to get reasonable statstcs from the fluctuatons one would have to run much larger systems for much longer tmes. ertanly a set of three smulatons could be set u n a sngle LAMMP nut fle to evaluate these roertes va the centered-fnte dfference aroach. hermodynamc relatons can also be exloted to evaluate other fluctuaton roertes. For examle the constant volume heat caacty s related to the above three roertes va v v (6.a) β and the sentroc comressblty can be obtaned from the followng relaton v β (6.b) β s III.. Proertes ncludng Entroy entroy here are many reasons to want to know the entroy of a system. here exst some aroxmate ways to evaluate an entroy based on an average confguraton. ee for examle the followng two references. D.. Wallace On the role of densty-fluctuatons n the entroy of a flud J. hem. Phys. 87 (4) (987) 8 84. A. Baranya D.J. Evans Drect entroy calculaton from comuter smulaton of lquds Phys. Rev. A 40 (7) (989) 87 8 However these efforts are fraught wth suffcent dffculty and errors that they have not been acceted as a standard way to comute entroy for arbtrarly comlex systems as s the case for the nstantaneous thermodynamc roertes. herefore today there s no easly mlemented and straghtforward way to calculate the entroy n a classcal smulaton. 8
D. Keffer ME 64 Det. of Materals cence & Engneerng Unversty of ennessee Knoxvlle 9 free energes A and G nce the entroy s not easly avalable there s also no drect way to calculate ether the Helmholtz or the Gbbs free energy from a smulaton. U A (7.a) U H G + (7.b) You should be suscous of any artcle that clams to have done so. chemcal otental of comonent µ he chemcal otental of comonent s defned as a thermodynamc artal dervatve j j A G µ (8) here are some sohstcated ways to comute the chemcal otental n a smulaton. We leave these technques to a searate lecture to be resented later n the course. entroy dfferences hose of you wth a fondness for thermodynamcs may recall Maxwell relatons. wo Maxwell relatons most relevant to the calculaton of entroy dfferences are (9.a) (9.b) hese equatons can be manulated and ntegrated to yeld ( ) ( ) d d (0.a) ( ) ( ) d d (0.b)
D. Keffer ME 64 Det. of Materals cence & Engneerng Unversty of ennessee Knoxvlle In the frst case one can erform a seres of smulatons all at the same temerature n whch the volume vares from to. One can then comute the value of across ths range usng for examle the centered-fnte dfference aroach descrbed above for each smulaton at ( ). From ths data one can use a numercal ntegraton scheme (such as the traezodal rule) to calculate an aroxmate value of the entroy dfference. ote that ths requres a seres of smulatons. In the second case one can erform a seres of smulatons all at the same temerature n whch the ressure vares from to. One can then comute the value of across ths range usng for examle the centered-fnte dfference aroach descrbed above for at each smulaton ( ). From ths data one can agan use a numercal ntegraton scheme (such as the traezodal rule) to calculate an aroxmate value of the entroy dfference. ote that ths requres a seres of smulatons. Other thermodynamc relatons can also be exloted to calculate entroy dfferences such as v (.a) (.b) As before these equatons can be manulated and ntegrated to yeld ( ) v d d (.a) ( ) ( ) ( ) d d (.b) Agan a seres of smulatons can be run ether at constant volume or constant ressure from to. If the arorate heat caacty s evaluated for each smulaton then the ntegral yelds the change n entroy. Once the change n entroy s known the change n free energy can be comuted drectly from equaton (7) snce the same smulatons generated values for the nternal energy and enthaly. One (of many) examles where such an ntegraton s done (usng equaton.a) s gven n Fgure.a. and Fgure.b of the followng reference. Mcutt.W. Wang Q. Keffer D.J. Ros O. Entroy-drven tructure and Dynamcs n arbon anocrystalltes J. anoartcle Res. 6 (4) 04 artcle # 65 do: 0.007/s05-04-65-7. 0