Nanoparticle Collisions in the Gas Phase in the Presence of
Singular Contact Potentials
Hui Ouyang, Ranganathan Gopalakrishnan, & Christopher J. Hogan Jr.*
Department of Mechanical Engineering, University of Minnesota, Minneapolis, MN, USA
Supplementary Information
Information Available:
-Regression equations for the enhancement factors in the continuum and free molecular
regimes for the non-retarded van der Waals and image potentials.
-The dimensionless equations of motion used in mean first passage time calculations.
- Tables of enhancement factors values in the continuum and free molecular regimes for
the non-retarded van der Waals (Table S1) and image (Table S2) potentials.
1. Regression Equations
With the dimensionless non-retarded van der Waals potential noted in equation (4a) of
the main text, with = 0.5the continuum enhancement factor (equation 3) is approximately
(within 1%) given by the expressions:
0.5704
 C  1  0.0586 VDW
0   VDW  4.5
(S1a)
1
4.5   VDW  300
C 
(S1b)
0.9976  0.0725 ln  VDW 
Similarly, the, the free molecular enhancement factor (evaluated using equations 6a and 6b) is
determined approximately from the expressions:
0.4202
 FM  1  0.3686 VDW
0   VDW  4.5
(S2a)
0.3913
 FM  1  0.3750 VDW
4.5   VDW  300
(S2b)
With the dimensionless image potential (equation 4b) of the main text, the continuum
enhancement factor is expressed approximately as:
 C  1  0.3475 I0.3802
0   I  266
(S3a)
0   I  266
0.50
 FM  1  1.2534 VDW
(S3b)
2. Equations of Motion


With v and r defined as the velocity and position vectors of the point mass, respectively, and


mij v


r
ai  a j  , the solution of the Langevin equation for the
f ij ai  a j 
point mass in the presence of van der Waals potentials can be written dimensionlessly as68:
2


^

*
*
*
*
2  FM
(S4a)
v      v   exp      VDW FVDW r ,  Kn D 2 1  exp    r  A1
C
with v * 
as well as r * 






2
*
r *      r *     v *      v *    2 VDW FVDW
r * ,  Kn 2D FM
 C2



^  1  exp    

r 
 1  exp    
(S4b)
2
^

 FM
r



A2
 C2
where  is the nondimensionalized time ( = (fij/mij)t) and  is small change in ; thus


*
  VDW FVDW
r * ,  Kn 2D




v*     , x *     , v *   , and x*   denote the point mass dimensionless velocities and
^
positions at dimensionless times  and  + ,respectively. The vector r and the scalar r* denote
a unit vector in radial direction and the dimensionless distance of the point mass from the origin


in the simulation domain, respectively. A1 and A 2 are both Gaussian distributed random vectors
with zero mean and variances given in equations (S4c) and (S4d), respectively:

2
2
(S4c)
1  exp  2 
A1  3Kn 2D FM
 C2

A2
2

 1  exp     
 FM


 



2
(S4d)
 C2 
 1  exp     
is the dimensionless van der Waals force between colliding entities, which is
 6Kn 2D
2
Finally, F*VDW
expressed as:


*
FVDW
r* , 
1  4 1   r *
4 1   r *


2
6  r *2  1 2
r *2  2  1





 r
2
8 1   r *
*2

 1 r *2  2  1
2




(S4e)
For the image potential, the dimensionless equations of motion of are similarly expressed as:

 

v *      v *   exp      I FI* r * Kn 2D
2
^

 FM




1

exp



r

A1
 C2




2
r *      r *     v *      v *    2 I FI* r * Kn 2D FM
 C2

 
 
  I FI* r * Kn 2D
^  1  exp    

r 
 1  exp    
(S5a)
(S5b)
2
^

 FM
r   A 2
2
C
and the dimensionless image force, F*I is expressed as:
  2r  1
r  1 r
FI* r * 
*2
*2
2
*3
All other parameters in equations (S4) and (S5) are defined in the main text.
(S5c)
Table S1. Summary of the enhancement factors in the continuum and free molecular regime for
the non-retarded van der Waals potential for the collision of two equal sized particles.
Y VDW  C  FM
Y VDW  C  FM
Y VDW  C  FM
Y VDW  C  FM
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
7.5
8
8.5
9
9.5
10
10.5
11
11.5
12
12.5
13
13.5
14
14.5
15
15.5
16
16.5
17
17.5
18
18.5
19
19.5
20
20.5
21
21.5
22
22.5
23
23.5
24
24.5
25
1.039
1.059
1.075
1.088
1.100
1.110
1.119
1.128
1.136
1.143
1.150
1.156
1.163
1.169
1.174
1.180
1.185
1.190
1.195
1.199
1.204
1.208
1.212
1.216
1.220
1.224
1.228
1.232
1.235
1.239
1.242
1.246
1.249
1.252
1.255
1.258
1.261
1.264
1.267
1.270
1.273
1.276
1.279
1.281
1.284
1.286
1.289
1.292
1.294
1.296
1.274
1.367
1.434
1.489
1.535
1.576
1.612
1.646
1.676
1.705
1.732
1.758
1.782
1.805
1.826
1.847
1.868
1.887
1.906
1.924
1.941
1.958
1.975
1.991
2.007
2.022
2.037
2.051
2.065
2.079
2.093
2.106
2.119
2.132
2.144
2.157
2.169
2.181
2.192
2.204
2.215
2.226
2.237
2.248
2.258
2.269
2.279
2.289
2.299
2.309
25.5
26
26.5
27
27.5
28
28.5
29
29.5
30
30.5
31
31.5
32
32.5
33
33.5
34
34.5
35
35.5
36
36.5
37
37.5
38
38.5
39
39.5
40
40.5
41
41.5
42
42.5
43
43.5
44
44.5
45
45.5
46
46.5
47
47.5
48
48.5
49
49.5
50
1.299
1.301
1.304
1.306
1.308
1.310
1.313
1.315
1.317
1.319
1.321
1.323
1.325
1.328
1.330
1.332
1.334
1.335
1.337
1.339
1.341
1.343
1.345
1.347
1.349
1.350
1.352
1.354
1.356
1.358
1.359
1.361
1.363
1.364
1.366
1.368
1.369
1.371
1.372
1.374
1.376
1.377
1.379
1.380
1.382
1.383
1.385
1.386
1.388
1.389
2.319
2.329
2.338
2.348
2.357
2.366
2.375
2.384
2.393
2.402
2.411
2.419
2.428
2.436
2.444
2.453
2.461
2.471
2.479
2.487
2.495
2.503
2.511
2.518
2.526
2.533
2.541
2.548
2.556
2.563
2.570
2.577
2.584
2.591
2.598
2.605
2.612
2.619
2.626
2.633
2.639
2.646
2.653
2.659
2.666
2.672
2.679
2.685
2.691
2.698
50.5
51
51.5
52
52.5
53
53.5
54
54.5
55
55.5
56
56.5
57
57.5
58
58.5
59
59.5
60
60.5
61
61.5
62
62.5
63
63.5
64
64.5
65
65.5
66
66.5
67
67.5
68
68.5
69
69.5
70
70.5
71
71.5
72
72.5
73
73.5
74
74.5
75
1.391
1.392
1.394
1.395
1.397
1.398
1.399
1.401
1.402
1.404
1.405
1.406
1.408
1.409
1.410
1.412
1.413
1.414
1.416
1.417
1.418
1.419
1.421
1.422
1.423
1.424
1.426
1.427
1.428
1.429
1.430
1.432
1.433
1.434
1.435
1.436
1.438
1.439
1.440
1.441
1.442
1.443
1.444
1.446
1.447
1.448
1.449
1.450
1.451
1.452
2.704
2.710
2.716
2.722
2.728
2.734
2.741
2.746
2.752
2.758
2.764
2.770
2.776
2.782
2.787
2.793
2.799
2.805
2.810
2.816
2.821
2.827
2.832
2.838
2.843
2.849
2.854
2.859
2.865
2.870
2.875
2.881
2.886
2.891
2.896
2.901
2.907
2.912
2.917
2.922
2.927
2.932
2.937
2.942
2.947
2.952
2.957
2.962
2.966
2.971
75.5
76
76.5
77
77.5
78
78.5
79
79.5
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
110
120
130
140
150
160
170
180
190
200
210
220
230
240
250
260
270
280
290
1.453
1.454
1.455
1.456
1.458
1.459
1.460
1.461
1.462
1.463
1.465
1.467
1.469
1.471
1.473
1.475
1.477
1.479
1.481
1.483
1.484
1.486
1.488
1.490
1.492
1.494
1.495
1.497
1.499
1.501
1.518
1.533
1.548
1.562
1.575
1.588
1.600
1.611
1.622
1.632
1.642
1.652
1.661
1.670
1.679
1.688
1.696
1.704
1.712
2.976
2.981
2.986
2.990
2.995
3.000
3.005
3.009
3.014
3.019
3.028
3.037
3.046
3.055
3.064
3.073
3.082
3.091
3.099
3.108
3.117
3.125
3.134
3.142
3.150
3.159
3.167
3.175
3.183
3.191
3.270
3.344
3.414
3.481
3.545
3.607
3.666
3.723
3.779
3.833
3.885
3.936
3.986
4.035
4.082
4.128
4.174
4.219
4.262
Table S2. Summary of the enhancement factors in the continuum and free molecular regime for
the image potential for the collision an uncharged particle with a point mass ion.
YI
C
 FM
YI
C
 FM
YI
C
 FM
YI
C
 FM
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
7.5
8
8.5
9
9.5
10
10.5
11
11.5
12
12.5
13
13.5
14
14.5
15
15.5
16
16.5
17
17.5
18
18.5
19
19.5
20
20.5
21
21.5
22
22.5
23
23.5
24
24.5
25
1.181
1.271
1.338
1.394
1.441
1.483
1.521
1.556
1.588
1.618
1.646
1.672
1.697
1.720
1.743
1.764
1.785
1.805
1.824
1.843
1.860
1.878
1.894
1.911
1.926
1.942
1.957
1.971
1.985
1.999
2.013
2.026
2.039
2.052
2.064
2.076
2.088
2.100
2.112
2.123
2.134
2.145
2.156
2.166
2.177
2.187
2.197
2.207
2.217
2.227
1.886
2.253
2.535
2.773
2.982
3.171
3.345
3.507
3.659
3.803
3.939
4.070
4.195
4.316
4.432
4.545
4.654
4.760
4.863
4.963
5.061
5.157
5.250
5.342
5.431
5.519
5.605
5.690
5.773
5.854
5.934
6.013
6.091
6.168
6.243
6.317
6.391
6.463
6.535
6.605
6.675
6.744
6.811
6.879
6.945
7.011
7.076
7.140
7.204
7.267
25.5
26
26.5
27
27.5
28
28.5
29
29.5
30
30.5
31
31.5
32
32.5
33
33.5
34
34.5
35
35.5
36
36.5
37
37.5
38
38.5
39
39.5
40
40.5
41
41.5
42
42.5
43
43.5
44
44.5
45
45.5
46
46.5
47
47.5
48
48.5
49
49.5
50
2.236
2.245
2.255
2.264
2.273
2.282
2.291
2.299
2.308
2.316
2.325
2.333
2.341
2.349
2.357
2.365
2.373
2.381
2.388
2.396
2.403
2.411
2.418
2.426
2.433
2.440
2.447
2.454
2.461
2.468
2.475
2.481
2.488
2.495
2.501
2.508
2.514
2.521
2.527
2.534
2.540
2.546
2.552
2.558
2.564
2.570
2.576
2.582
2.588
2.594
7.329
7.391
7.452
7.513
7.573
7.632
7.691
7.749
7.807
7.865
7.922
7.978
8.034
8.090
8.145
8.200
8.254
8.308
8.362
8.415
8.468
8.520
8.572
8.624
8.675
8.726
8.777
8.827
8.877
8.927
8.976
9.025
9.074
9.122
9.171
9.219
9.266
9.314
9.361
9.408
9.454
9.500
9.547
9.592
9.638
9.683
9.728
9.773
9.818
9.862
50.5
51
51.5
52
52.5
53
53.5
54
54.5
55
55.5
56
56.5
57
57.5
58
58.5
59
59.5
60
60.5
61
61.5
62
62.5
63
63.5
64
64.5
65
65.5
66
66.5
67
67.5
68
68.5
69
69.5
70
70.5
71
71.5
72
72.5
73
73.5
74
74.5
75
2.600
2.606
2.612
2.617
2.623
2.628
2.634
2.640
2.645
2.651
2.656
2.661
2.667
2.672
2.677
2.683
2.688
2.693
2.698
2.703
2.709
2.714
2.719
2.724
2.729
2.734
2.739
2.743
2.748
2.753
2.758
2.763
2.768
2.772
2.777
2.782
2.786
2.791
2.796
2.800
2.805
2.809
2.814
2.818
2.823
2.827
2.832
2.836
2.840
2.845
9.907
9.951
9.994
10.038
10.081
10.124
10.167
10.210
10.253
10.295
10.337
10.379
10.421
10.462
10.504
10.545
10.586
10.627
10.668
10.708
10.749
10.789
10.829
10.869
10.908
10.948
10.987
11.027
11.066
11.105
11.143
11.182
11.221
11.259
11.297
11.335
11.373
11.411
11.449
11.486
11.523
11.561
11.598
11.635
11.672
11.708
11.745
11.781
11.818
11.854
75.5
76
76.5
77
77.5
78
78.5
79
79.5
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
110
120
130
140
150
160
170
180
190
200
210
220
230
240
250
260
2.849
2.853
2.858
2.862
2.866
2.871
2.875
2.879
2.883
2.887
2.896
2.904
2.912
2.920
2.928
2.936
2.944
2.952
2.959
2.967
2.974
2.982
2.989
2.997
3.004
3.012
3.019
3.026
3.033
3.040
3.108
3.172
3.232
3.289
3.343
3.394
3.443
3.490
3.535
3.578
3.620
3.660
3.699
3.737
3.773
3.809
11.890
11.926
11.962
11.998
12.033
12.069
12.104
12.140
12.175
12.210
12.280
12.349
12.418
12.487
12.555
12.623
12.690
12.757
12.824
12.890
12.956
13.021
13.087
13.151
13.216
13.280
13.344
13.407
13.470
13.533
14.145
14.729
15.290
15.829
16.350
16.853
17.341
17.815
18.276
18.725
19.162
19.590
20.007
20.416
20.817
21.209