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Hard Sphere Thermodynamic and Transport Properties

BULK-SLIT-HS: properties of the hard-sphere fluid in bulk and confined to a slit pore 
This page gives some results for hard-sphere fluids [1]

Methods: 
Event-driven molecular dynamics (MD) [1] and Transition-matrix Monte Carlo (TMMC) [2].

Remarks: 

TMMC data is taken from Refs.[5,6].  Data for MD simulations was produced using N=4000 particles. All systems were equilibrated for 1000 reduced time units. Productions runs of time t>100,000 reduced time units were used. Self-diffusivities were calculated from the Einstein relation D_x=dx(t)^2, t->infinity, were dx(t)^2 is the mean square displacement in direction x. For the bulk self-diffusivity, all three directions where averaged together.  For the slit-pore self-diffusivity, only the periodic x and y directions (see schematic below) were averaged together. Mean-squared displacements were measured over times windows exceeding 5000 reduced units, using standard block averaging [1]. No less than 5 independent (different random seeds) were conducted at each state point.  This data was averaged to give the mean properties.  Errors were estimated by standard methods to yield the 95% confidence intervals.  That is, the error was take as T(n)*STD/sqrt(n), where n is the number of independent samples, T is the inverse student's t PDF at 95% level for n samples, and STD is the sample standard deviation of the independent samples.

 

Bulk properties:
sigmaparticle diameter
mparticle mass
betainverse temperature
densdensity (reduced by sigma^3)
mureduced chemical potential
sxreduced excess entropy (relative to ideal gas at same temperature and density)
P_MCreduced pressure from TMMC simulations
P_MDreduced pressure from MD simulations
Dreduced self-diffusion from MD simulations
+/-95% confidence intervals for data in preceding column

 

Data:

 

densmusxP_MCP_MD+/-D+/-
0.025-3.581518-0.0512390.0263520.0263460.0000008.3383230.013053
0.050-2.775580-0.1062670.0555930.0555820.0000004.0648060.006030
0.075-2.251420-0.1632170.0880230.0880050.0000022.6405030.002924
0.100-1.838804-0.2221350.1239690.1239420.0000011.9345990.003292
0.125-1.483998-0.2831460.1637950.1637560.0000021.5134580.003706
0.150-1.163011-0.3463500.2078770.2078450.0000031.2326590.002354
0.175-0.862397-0.4118570.2566940.2566460.0000041.0330160.002243
0.200-0.574198-0.4798050.3107130.3106490.0000060.8851750.001139
0.225-0.292855-0.5503360.3704870.3703980.0000060.7704230.000901
0.250-0.014585-0.6235750.4365740.4364780.0000070.6794250.000883
0.2750.263858-0.6996790.5096680.5095530.0000100.6038510.001265
0.3000.544764-0.7788020.5904370.5903540.0000070.5412490.000667
0.3250.830785-0.8611310.6798290.6796870.0000130.4878710.000904
0.3501.123470-0.9468500.7786270.7784770.0000130.4419600.000725
0.3751.424630-1.0361600.8878180.8876790.0000200.4015390.000660
0.4001.736449-1.1292861.0086711.0084520.0000240.3655410.000291
0.4252.060345-1.2264641.1423051.1420620.0000210.3336210.000941
0.4502.398216-1.3279471.2901551.2898580.0000310.3046740.000569
0.4752.751827-1.4340121.4537351.4533880.0000220.2780820.000337
0.5003.122964-1.5449491.6347021.6344670.0000250.2538730.000257
0.5253.514262-1.6610791.8352871.8349800.0000390.2319950.000359
0.5503.927635-1.7827682.0575222.0571190.0000460.2112710.000357
0.5754.364901-1.9103882.3035342.3034480.0000180.1920910.000523
0.6004.829816-2.0443312.5767352.5766400.0000310.1747330.000458
0.6255.324387-2.1850972.8797142.8799720.0000430.1579140.000271
0.6505.853256-2.3331293.2169513.2168750.0000670.1425690.000238
0.6756.415393-2.4889993.5894343.5915270.0000460.1283860.000154
0.7007.021445-2.6533144.0061374.0084620.0000540.1147390.000185
0.7257.676348-2.8266294.4729144.4728770.0000550.1024000.000191
0.7508.387102-3.0100794.9972084.9907870.0000250.0903960.000136
0.7759.139130-3.2040615.5708495.5691150.0000900.0793760.000099
0.8009.951519-3.4094166.2107716.2155510.0000970.0689040.000098
0.82510.849336-3.6271956.9404876.9390640.0000990.0593160.000062
0.85011.822939-3.8592027.7558007.7502630.0000960.0503060.000035
0.87512.868237-4.1055968.6575788.6610190.0000630.0419380.000062
0.90014.019667-4.3678029.6798689.6856150.0001200.0343270.000087

 

Slit Pore Properties

img19

 

 

 

 

Figure 1: Schematic of slit-pore geometry. Wall separation is given by H, and center accessible separation given by h=H-sigma, where sigma is the particle diameter.

Data:

HWall separation
sx_1Excess entropy relative to an ideal gas with a  density profile equivalent to the actual fluid
sx_2Excess entropy relative to an ideal gas with a  uniform density profile
P_MC_TPore averaged transverse pressure calculated from TMMC  simulations [3]
P_MD_TPore averaged transverse pressure calculated  from molecular dynamics simulations (see  [4])
P_MD_NNormal pressure calculated from MD simulations
P_bulk_muEquilibrium pressure of the bulk fluid (that  is, the pressure of the bulk fluid at the same value of chemical potential)


Note: NaN implies no data available at that state point.

 

 

 

Gapdensmusx_1sx_2P_MC_TP_MD_T+/-P_MD_N+/-D+/-
5.00.05-2.523062-0.119700-0.1204030.0704280.0704310.0000010.0695910.0000013.6108980.007756
5.00.10-1.553323-0.248980-0.2519140.1590040.1590090.0000020.1553140.0000031.7533380.004868
5.00.15-0.840555-0.386410-0.3929800.2696910.2696910.0000040.2605410.0000081.1412610.002083
5.00.20-0.210475-0.532890-0.5446590.4072860.4073050.0000130.3893420.0000110.8348700.001844
5.00.250.395759-0.689490-0.7080580.5777550.5777370.0000120.5467070.0000110.6476950.001386
5.00.301.007739-0.857400-0.8846710.7882140.7882130.0000170.7386460.0000190.5217830.002015
5.00.351.645818-1.037980-1.0759131.0476151.0476150.0000180.9725470.0000130.4270860.001786
5.00.402.326486-1.232900-1.2837561.3669411.3670780.0000311.2576040.0000450.3531180.000642
5.00.453.066765-1.443920-1.5101751.7605691.7604720.0000411.6057490.0000370.2928930.000731
5.00.503.880649-1.673040-1.7576862.2442422.2448630.0000312.0326410.0000470.2423930.000643
5.00.554.787536-1.921920-2.0291682.8398912.8405050.0000252.5584100.0000440.1983150.000530
5.00.605.803869-2.191380-2.3277453.5710093.5709980.0000873.2062490.0000900.1613600.000294
5.00.656.946580-2.482570-2.6566984.4644884.4648640.0000843.9975960.0000740.1289250.000389
5.00.708.248008-2.797450-3.0199135.5636065.5626120.0000694.9456320.0001110.1005380.000249
5.00.759.762154-3.140440-3.4236206.9371126.9297740.0000866.0543810.0001100.0752430.000242
5.00.8011.549069-3.523940-3.8751608.6700228.6772270.0000507.3281060.0001860.0531360.000215
5.00.8513.741602-3.963690-4.38648810.93230710.9809520.0001268.7966090.0002460.0341850.000193
5.00.9014.389045NaNNaN11.632706NaNNaNNaNNaNNaNNaN
6.00.05-2.568841-0.117344-0.1175990.0674650.0674590.0000010.0668360.0000023.6825240.007993
6.00.10-1.604890-0.244385-0.2465020.1519780.1519780.0000040.1492270.0000031.7782120.003673
6.00.15-0.898488-0.379640-0.3847100.2572770.2572720.0000050.2504290.0000051.1510700.002664
6.00.20-0.275511-0.523986-0.5332620.3878720.3878770.0000080.3744010.0000050.8380720.002677
6.00.250.322632-0.678540-0.6933380.5493370.5493120.0000140.5259590.0000130.6495360.001188
6.00.300.925286-0.844455-0.8662470.7482890.7483490.0000120.7109900.0000120.5221840.000409
6.00.351.553131-1.023098-1.0535040.9933130.9933240.0000150.9366940.0000160.4272010.000716
6.00.402.222081-1.216020-1.2569051.2945821.2946020.0000311.2119770.0000270.3526090.000968
6.00.452.947651-1.424940-1.4784431.6649521.6650020.0000261.5478890.0000600.2930250.000418
6.00.503.746670-1.651940-1.7205992.1208192.1206470.0000661.9585870.0000500.2424760.000884
6.00.554.635354-1.899227-1.9861612.6811912.6814610.0000572.4621370.0000280.1996690.000649
6.00.605.633863-2.168980-2.2782403.3707603.3717770.0000973.0813070.0000710.1626780.000824
6.00.656.767947-2.463380-2.6009434.2220754.2214580.0000483.8418140.0000770.1302510.000312
6.00.708.059413-2.784407-2.9581805.2691355.2696210.0000774.7689280.0000910.1019800.000262
6.00.759.559559-3.136967-3.3554186.5757156.5743290.0000965.8861060.0001080.0771130.000336
6.00.8011.330729-3.530512-3.8005288.2244438.2272100.0001237.2265470.0001630.0556670.000154
6.00.8513.500937-3.981122-4.30457910.37528910.3659510.0001308.8768450.0004320.0370100.000108
6.00.9016.505228-4.504592NaN13.642548NaNNaNNaNNaNNaNNaN
7.00.05-2.600641-0.115640-0.1159390.0654780.0654790.0000010.0649860.0000013.7299690.011858
7.00.10-1.640697-0.241170-0.2429250.1472920.1472950.0000010.1451140.0000041.7963640.003717
7.00.15-0.938752-0.374691-0.3790670.2490130.2490040.0000040.2435680.0000051.1603070.001509
7.00.20-0.320857-0.517648-0.5253800.3749430.3749420.0000060.3642130.0000100.8427510.001596
7.00.250.271487-0.670559-0.6830010.5303970.5303890.0000180.5117600.0000180.6504260.001899
7.00.300.868022-0.835136-0.8533050.7218610.7218480.0000130.6919810.0000260.5208770.001007
7.00.351.488327-1.012263-1.0377940.9572120.9572700.0000130.9119120.0000250.4273740.000729
7.00.402.149188-1.203859-1.2380731.2465691.2465170.0000431.1803100.0000420.3525270.001370
7.00.452.864824-1.411375-1.4562401.6017081.6019070.0000291.5079770.0000340.2928580.000549
7.00.503.652093-1.637099-1.6946152.0383702.0386430.0000321.9084380.0000430.2430900.000211
7.00.554.529029-1.883148-1.9560632.5759752.5761090.0000572.3988740.0000520.2003230.000245
7.00.605.513861-2.152178-2.2435753.2372303.2381690.0000963.0009840.0000730.1631850.000391
7.00.656.632302-2.446995-2.5613004.0534774.0548120.0000483.7415820.0000630.1313790.000405
7.00.707.912163-2.769997-2.9132335.0621235.0642990.0000674.6510180.0000990.1034010.000315
7.00.759.391189-3.127170-3.3052406.3143966.3196170.0000745.7633420.0001310.0788440.000277
7.00.8011.149337-3.523552-3.7443257.9055017.8976760.0001217.1262590.0001610.0574800.000170
7.00.8513.224048-3.972839-4.2404749.9044329.9011270.0001928.8405160.0002390.0391140.000097
7.00.9015.630033-4.476738NaN12.361201NaNNaNNaNNaNNaNNaN
7.50.05NaNNaNNaNNaN0.0647160.0000010.0642720.0000013.7499980.007295
7.50.10NaNNaNNaNNaN0.1454930.0000040.1435220.0000041.8042810.004415
7.50.15NaNNaNNaNNaN0.2458230.0000060.2408980.0000051.1615730.001582
7.50.20NaNNaNNaNNaN0.3699790.0000060.3602500.0000060.8440420.001627
7.50.25NaNNaNNaNNaN0.5231280.0000130.5062260.0000100.6520810.001642
7.50.30NaNNaNNaNNaN0.7116730.0000200.6845510.0000170.5223220.001894
7.50.35NaNNaNNaNNaN0.9434290.0000280.9022240.0000350.4272070.001074
7.50.40NaNNaNNaNNaN1.2280910.0000321.1678940.0000480.3537780.001053
7.50.45NaNNaNNaNNaN1.5776730.0000381.4922940.0000290.2932380.000931
7.50.50NaNNaNNaNNaN2.0072240.0000221.8888480.0000590.2433920.000585
7.50.55NaNNaNNaNNaN2.5356040.0000752.3741420.0000630.2008600.000535
7.50.60NaNNaNNaNNaN3.1867880.0000932.9691760.0000750.1641730.000511
7.50.65NaNNaNNaNNaN3.9913700.0000963.7006300.0000940.1318820.000266
7.50.70NaNNaNNaNNaN4.9884540.0000754.6051410.0001190.1040810.000171
7.50.75NaNNaNNaNNaN6.2240150.0000925.7315210.0000890.0794560.000155
7.50.80NaNNaNNaNNaN7.7491190.0000627.1358380.0001240.0582100.000159
7.50.85NaNNaNNaNNaN9.6224710.0001508.8483160.0002340.0399230.000186
8.00.05-2.624020-0.114500-0.1149050.0640610.0640640.0000010.0636580.0000003.7702500.006176
8.00.10-1.666879-0.238670-0.2403840.1439650.1439520.0000020.1421510.0000021.8122980.002364
8.00.15-0.968441-0.371160-0.3749840.2431090.2431070.0000020.2386100.0000081.1661240.002646
8.00.20-0.354264-0.512830-0.5196300.3657210.3657190.0000070.3568300.0000080.8458260.001143
8.00.250.233732-0.664730-0.6754950.5168800.5169070.0000080.5014500.0000160.6515500.001909
8.00.300.825575-0.828070-0.8438010.7029550.7029540.0000120.6781400.0000230.5234220.001404
8.00.351.440646-1.004180-1.0260540.9315640.9315540.0000230.8938240.0000310.4279720.000715
8.00.402.095303-1.194660-1.2241171.2123571.2122680.0000171.1571270.0000290.3538690.000868
8.00.452.804028-1.401210-1.4397531.5568981.5569560.0000361.4786880.0000240.2939080.000649
8.00.503.583470-1.625860-1.6753631.9804041.9803990.0000511.8717620.0000480.2433570.000803
8.00.554.450287-1.871010-1.9337412.5009472.5012510.0000472.3531290.0000590.2015290.000339
8.00.605.425181-2.139340-2.2179993.1421403.1428350.0000702.9438260.0001110.1645200.000314
8.00.656.531029-2.433780-2.5318233.9327513.9346960.0000443.6705950.0001080.1326600.000258
8.00.707.804626-2.758190-2.8802614.9160974.9143200.0000624.5657290.0001120.1047540.000187
8.00.759.269163-3.116060-3.2682626.1305746.1315740.0001065.6679440.0001090.0800810.000204
8.00.8010.987243-3.513730-3.7020527.6538687.6544250.0001557.0308240.0000820.0587940.000219
8.00.8513.000985-3.959060-4.1912999.5538409.5659890.0001698.7480160.0003130.0405380.000121
8.00.9015.385838-4.447830NaN11.940553NaNNaNNaNNaNNaNNaN
9.00.05-2.641729-0.113442-0.1134230.0630060.0630030.0000010.0626580.0000013.8008160.006367
9.00.10-1.687112-0.236785-0.2378750.1414490.1414440.0000040.1399140.0000031.8234100.003640
9.00.15-0.991100-0.368236-0.3713620.2387000.2386840.0000030.2348570.0000071.1720990.002518
9.00.20-0.379936-0.509107-0.5147760.3588040.3588040.0000080.3512360.0000090.8487480.002412
9.00.250.205031-0.660059-0.6692480.5068340.5068060.0000070.4936210.0000150.6546820.002184
9.00.300.793081-0.822592-0.8361320.6888300.6888000.0000100.6676060.0000290.5240300.000874
9.00.351.403831-0.997799-1.0168170.9122810.9123190.0000230.8800520.0000330.4285530.000666
9.00.402.053439-1.187455-1.2130371.1865521.1866460.0000131.1394610.0000380.3540970.000679
9.00.452.757311-1.393174-1.4267361.5233811.5233470.0000211.4563050.0000350.2944090.000642
9.00.503.530144-1.617063-1.6602791.9367321.9368800.0000281.8437320.0000280.2446140.000428
9.00.554.390853-1.861504-1.9163302.4455412.4452630.0000462.3181970.0000640.2020470.000255
9.00.605.358016-2.129296-2.1980933.0717313.0714910.0000602.9005250.0000640.1653150.000403
9.00.656.454986-2.423515-2.5091073.8437363.8444300.0000703.6169910.0000570.1335910.000196
9.00.707.712832-2.747473-2.8541804.7997654.8009680.0001124.5006360.0000430.1057090.000255
9.00.759.168933-3.106304-3.2387235.9885165.9889330.0001115.5921640.0001230.0812200.000180
9.00.8010.874549-3.503939-3.6681757.4774417.4711330.0000676.9463460.0001080.0601020.000165
9.00.8512.862319-3.946623-4.1516199.3242329.3231680.0000578.6470430.0002620.0416930.000096
9.00.9015.158590-4.429011NaN11.585506NaNNaNNaNNaNNaNNaN
10.00.05-2.655929-0.112760-0.1131160.0621750.0621790.0000010.0618800.0000013.8280650.008916
10.00.10-1.702996-0.235300-0.2366570.1395010.1394980.0000030.1381680.0000031.8346710.004733
10.00.15-1.009155-0.366100-0.3690870.2352540.2352480.0000040.2319170.0000081.1772620.003609
10.00.20-0.400188-0.506140-0.5114350.3534450.3534380.0000050.3468480.0000050.8501340.002086
10.00.250.182100-0.656430-0.6648170.4989730.4989630.0000180.4874740.0000130.6555630.001497
10.00.300.767246-0.818180-0.8304570.6778320.6778060.0000180.6593420.0000120.5253340.000744
10.00.351.374626-0.992750-1.0098190.8973080.8973570.0000170.8692230.0000170.4291750.000592
10.00.402.020643-1.181660-1.2046011.1666921.1667190.0000231.1255370.0000270.3549840.000341
10.00.452.720036-1.386730-1.4168041.4972431.4972640.0000221.4386900.0000480.2954040.000475
10.00.503.488151-1.609940-1.6486161.9030061.9030220.0000381.8215910.0000590.2456250.000558
10.00.554.342587-1.853810-1.9027082.4018612.4018530.0000312.2907080.0000560.2032960.000567
10.00.605.302675-2.120990-2.1822863.0158063.0160960.0000752.8664330.0000770.1663960.000239
10.00.656.392146-2.414780-2.4911183.7730393.7742900.0000333.5749530.0000380.1345620.000186
10.00.707.641344-2.738850-2.8336604.7106944.7124700.0001114.4491260.0000820.1066760.000187
10.00.759.079698-3.097300-3.2145875.8704725.8774420.0000775.5307020.0000630.0823470.000143
10.00.8010.760901-3.495590-3.6418817.3189747.3293990.0000676.8740780.0001470.0609090.000141
10.00.8512.736051-3.938110-4.1224079.1305069.1411040.0001178.5560130.0002130.0424950.000074
10.00.9015.013105-4.421070NaN11.344811NaNNaNNaNNaNNaNNaN
15.00.05-2.697185-0.110414-0.1104580.0598220.0598240.0000010.0596450.0000013.8995760.005454
15.00.10-1.749725-0.230747-0.2314390.1339450.1339400.0000020.1331410.0000041.8646680.003974
15.00.15-1.061974-0.359472-0.3611700.2254530.2254520.0000030.2234460.0000031.1923090.002397
15.00.20-0.459774-0.497250-0.5005280.3381520.3381320.0000060.3341540.0000060.8617870.001197
15.00.250.114696-0.645286-0.6506630.4765950.4765970.0000100.4696370.0000130.6632270.000896
15.00.300.691041-0.804982-0.8127760.6464670.6464930.0000060.6352750.0000140.5305810.000762
15.00.351.288918-0.977541-0.9883190.8547860.8547730.0000150.8376550.0000260.4336370.000593
15.00.401.923592-1.164250-1.1789631.1099861.1100190.0000261.0848960.0000290.3591060.000945
15.00.452.609898-1.367111-1.3865251.4227561.4229110.0000351.3871780.0000260.2992880.000430
15.00.503.363557-1.588511-1.6133171.8066531.8067440.0000461.7569720.0000400.2493530.000550
15.00.554.200814-1.830632-1.8618482.2780252.2782040.0000302.2101550.0000480.2069590.000337
15.00.605.141221-2.095960-2.1353572.8578842.8584470.0000222.7666190.0000730.1707050.000615
15.00.656.209108-2.388056-2.4373203.5736613.5742620.0000623.4517580.0000770.1385410.000317
15.00.707.431799-2.711255-2.7721344.4586944.4600000.0000354.2978870.0000970.1105000.000360
15.00.758.839178-3.070076-3.1452325.5526415.5595240.0000625.3464280.0000980.0860700.000297
15.00.8010.484866-3.469050-3.5630446.9200356.9293390.0000966.6503620.0001560.0645730.000276
15.00.8512.432600-3.912918-4.0322938.6430628.6417650.0001028.2780020.0000830.0458550.000036
15.00.9014.681943-4.408871NaN10.752998NaNNaNNaNNaNNaNNaN
Self-diffusion versus densitySelf-diffusion vs excess entropy type 1
fig2

self-diffusion vs excess entropy 1
Self-diffusion vs excess entropy type 2 
diffusion vs excess entropy 2
 
 

References

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2. J.R. Errington, J. Chem. Phys. 118, 9915-9925 (2003).

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4. J. R. Henderson, F. van Swol, Mol. Phys. 51, 991 (1984).

5. J. Mittal, J. R. Errington, T. M. Truskett, Phys. Rev. Lett. 96, 177804 (2006).

6. G. Goel, W. P. Krekelberg, M. J. Pond, J. Mittal, V. K. Shen, J. R. Errington, T. M. Truskett, J. Stat. Mech. 2009, P04006 (2009).

Created September 25, 2012, Updated August 25, 2016