- Email: [email protected]
Rotational diffusion tensors for non-spherical molecules
Volume 71, number
ROTATIONAL Donald
CHEhIlCAL
I
DIFFUSION
C. KNAUSS,
Department
TENSORS
PHYSICS
LETTERS
FOR NONSPHERICAL
1 Aprd 1980
MOLECULES
Glenn T. EVANS
of Chemurry. Oregon State Unwemty,
Con allis.
Oregon 9 733 I, USA
and Davxd M. GRANT Department
of Che.nirtry.
Received 5 December
1979,
Unhersqv
in foal
of Utah, Salt Luke City, Utah 84i1.2, USA form 15 January
1980
Tbc rotatlonal tifuslon tensors of cweral ngld molecules are determmed from the rotational fnctlon tensor. The molecular friction tensor IS modeled as a sum over the translational Stokes law fnctton coefficients of the constituent atoms of the molecule. Agreement of the theory w~tb expenmental rotatIonal tifusion constants and orlentattonal correlatton times Is surpramgly good.
1. Introduction Browman dynamxs simulations [l-7] of chain molecules have been apphed recently to the calculation of various transport coefficients of flevlble molecules m hquid solvents. At the crux of the brownian motion procedure is a friction constant for the atoms of the parent chain molecule. A pnon It IS not clear how to represent the fricuonal charactenstlcs of constituent atoms in a molecule. Mazo [8] has found that the translational diffusion coefficient of an alkane cham can be quantitatively determined by means of the Kirkwood formula parameterized m terms of the fnction coefficient per backbone atom. The actual fnction coefficient of the backbone atom represents the carbon atom by its covalent ra&us (0.77 A) and stick boundary conditions in the context of Stokes law. Evans and Knauss [7] have found that the dlelectnc correlation firne of a series of I-bromoalkanes can also be accommodated by the same parametenzatlon. With this in mind, it seemed mtngumg to apply the same prescnption to small rigd molecules, such as methylene habdes, CS2, and benzene, since the rotatlonal Mfusion tensors and orientational correlation tunes have been determined [9-l 3] _ The chosen model IS as follows: a rigd molecule is 158
represented by a collectlon of hydrodynamically noninteractmg spheres, fiuted at distances and angles consistent ~th the known molecular structure. The friction coefficient of each atom 1s gven by the translational Stokes law and the overall friction tensor IS then constructed from the appropnate tensor sums of fnctron constants. The molecular diffusion tensor 1s the mverse of the molecular friction tensor. In the following section, we present the theory underlymg this work and an application to a few simple rigid molecules.
2. Theory The specific distnbution function for a tagged molecule in a fluid shall be defined as 9(f), which is a function of all the coordmates of the atoms that constitute the molecule. \k(t) obeys a continuity equahon in Remann space [2,14-161
a,*(t)
= ~l/g1/2)(a/aQ’)[g112hi~(t)]
_
Q’ IS a generalized coordmate, al(r) the dynamical generabzed velocity, and g the covariant metric determinant. The covariant metnc tensor, G,, is
Volume 7 1, number 1
CHEMICAL PHYSICS LE-F-FBRS
3N
(2) where (3)
_
?he Set of d xk, {xk,
k = 1, 3N),
Comprk
the
x,
H = RSS_ Rsh . (Bhh)-1
-
In actual fact, the Qi shall be chosen to be the Euler angles, bond angles, and bond lengths of the molecule. To recast eq. (1) into a drffusion equation, we make use of the force balance equation,
,
(4)
with Fk = -_(a/axk)(kBT
h \k+ (/) .
(5)
Eqs. (4) and (5) relate the total force to the sum of the entropic and intramolecular (or torsional) forces. As wntten in eq. (4), the friction coefficient vector IS 3ZVdimensional and, in principle, one could allow for anisotropic translational friction of each of the atoms of the molecule. This wrll not be done, but rather
{‘Yk,k=1,3N3=Cy,,~=Yn,To,llb,~~,Yg,-.-3Next,
transform
the cartesian
velocity
(ala@ + B aujag+tqt)
to Riemann
space, then
. Bhs ,
(12)
of the “soft-soft” part of the covariant metric C. By defining moments of resistance in a mmner analogous to the moment of inertia (replacing mass by friction constant), we find that the elements of the symmetric covariant tensor S (the inverse of H) are simply related to these moments and the Euler angles. We pick a body-fuced coordinate system which diagonalizes the moment of resistance tensor (i.e., friction tensor) and fiid that the elements of S are [IS1
SOS= K y y sin2a + Kxx cos2cl,
+ K y ,r cos2a sin2j3+ Kzz
A[k = a&axk
(7)
and (8)
(9)
cos2j3 ,
Spr=(KXX-Kyy)cos~sinasinB, Sa7=KzzcosP,
k,T ln aI) _
As a result, the velocity in generalized coordmates is
and
(IV
s Y-Y = Kxx sin’s sin20
where
-(a/a&(u+
P= W,T,
RhS being the “hard-soft” part of the tensor R. Thus the implied sums in eq. (11) go over only the “‘soft” variables. For a heterogeneous fle_xible chain molecule, the soft vanables would consist of the thee Euler angles and the torsion angles (assuming that bond lengths and nearest neighbor bond angles are fxed)For rigid systems, the soft variables are only the tf.1122 Euler angles, and Q is just the distribution function of the Euler angles. We can show (see appendix 1) that the set of “hard” variables (combinations OF the bond lengths and bond angles) can be chosen such that Rhs = 0 and g = gs, where gs is the determinant
s P(L =Kzz,
fi =
,
where H is the contravariant tensor,
molecule. Thus, the xk denote the ordered set, Cq, ‘2, --- rrl
x
y,
and z Cartesian coordinates of the N vectors, r,, that specrfy the location of the N atoms comprisiig the
= +Fk
If we now implement constraints on the so-called “hard” variables in the usual manner [2,14-161, we find
a,@(t) = (kT/g’12)(W3Qi)g’12H~
Blk = axkjaQ’
&
1 ApilL
Sap=%
03)
with
H=S-’
_
04)
Q, p, and 7 are the Euler angles and are defined in accord wi*& our previous work [IS] _The K tensor is akin to an inertia tensor with the friction constant playing the role of mass; hence,
159
1 Aprit 1980
CHEMICAL PHYSICS LETTERS
Volume 71, number I N
K=C y (AL-rJrJ) J J
,cl
in the above context, we are associating an tsotroptc transiattonal frrctton constant wrth each of the con-
stituent N atoms. rather than wrth each of the 3X coordmates. Folfowmg earher work [ 151, one tnverts S to form a diffusron equation 2,8(f)
= r.I * D
l
+5’(t)
,
WI
IJ is the lab-fLsled rota&on operator rotatronal diffusron tensor, D,J =k,T
$J&,
and D is the
-
which IS given ehphcitly rn terms of the elements of the friction tensor, and consequently the frrctron constants of the constituent partrcles. Eq (15) doffers from Kirkwood’s diffuston tensor ordy rn as much as we have allowed the constrtuent atoms to have ddferent frictron constants.
3. Comparison
with experiment
3.1. Appkatron
to nzethylerre halides / CH2Xz)
To apply eq. (16), we must dtagonahze thi friction tensor. K. Consequently, define our body-fixed coordmate system for methy lene hahdes as follows- The 2 axis is parallel to the lute Jouung the two halogens, the Y aus IS parallel to the hne ~ommg the two hydrogcns. and the X axis is perpendrcular to both. Referrrng to fig_ 1. we see that the various atoms have pontion vectors
0. A),
rxt=(B-D, r&=(B+F, rc =(B,
-.t?, o),
rxz=(B-D, ‘H~=(B+F.
for the translational drag on the kth atom, yk = mLrkn% where v = 6(4) for stick(shp) boundary condrtrons and ak is the appropnate radius of the kth particle. then the elements of the fnction tensor become Kxxfvml
= 2(aHEr +aXA2),
K,,/znrq
= 2[aXA2
+aX(D
-B)’
+a~(B+l=)~]
w?,B2,
Kzzjvnq
= 2[aHE2 + aH(B + F)* + ax(D - B)2j
+a,B’,
Table 1 Geometnc parameters (angles in degxe, distances m A) used m the methytene halide cakulatlons
E, o),
0, 0).
X
Fag 1. Coordmate system for the methylene habdes.
0. -A).
Thus defined. the fnctron tensor is dragonal, mdependent of the choice of B. B IS chosen wrthout ambiguity as a uruque point m the body. vrnz.,tire center of resistance, which has the property of decoupling translatmnal and rotational motions (see appendtx 2) m the drffusroaal regrme It is noteworthy that the center of mass of the molecule does not enter the descrrption of the brownian dynanucs, but rather rts center of resistance [ 17,I 81. fmplementing the Stokes relation 160
“‘1
(16)
LX-C-X f. H-C-H IC-Xl E-HI
ax “C OH
C&$32
CHzBr2
(32
1118b) 106.9 1 772 b, 1.079 1.002 0.770 0 310
112 7 =) 105 9 1.927 =) 1.079 1.157 0.770 0 310
114.7 d) 103.6 2.12 d) 1.079 1.350 0 770 0 310
12
a) ar_-taken to be the covalent radms of carbon. OX, aH found from UC and the bond lengths The angle LH-C-H estxmated from the reauuement that tX-C-H retam the tetrahedral value b, Ref (211 c, Ref [221. d, Ref. [23].
Volume 71, number 1
CHEMICAL
PHYSICS
Table 2 Reduced dtffusion constants for methylene hahdes CHsC12
CHaBra
CH2 I2
Dirx
0 209
0 156
0111
D’YY
0.172
0.130
0 094
Diz
0 512
0 453
0 399
PX
0.408
PX(exp ) PY P y(exp.)
0 344
0 180 a) (0 166 - 0.199) 0.336 0 250 a) (0.242 - 0 258)
0 278 0205-~0003~)
0.287
0.236
-
0079=0057b)
b, Ref
[IO]
n (cP)
- a#)/[2(ax
+ aH) + ac]
-
The remaming parameters are readdy determined gven the bond angles and bond lengths. The parameters used are listed in table 1 to obtam the results of table 2, namely, the reduced drffusron constants
D:, = W-v/kTW,,
0°C
-4OT
-80°C
0.43
0.62
0.99
Dxx
X lOto s-l
9.7 (5.4)
5.7 (2.9)
3.0 (1.2)
Dyy
X 10”
8.1 (6.6)
4.8 (4.3)
2.5 (L-9)
Dzz
X 1O’O s-r
24 (27)
14 (17)
7-4 (7.5)
s-r
a) Expertmental values rn parentheses.
The second rank orientational correlation time, ‘2, is related to D, in the brownian limit by 72 = l/60,
where B = 2(f7,D
Table 3 Calculated and experimental diffusion constants for CDaCLa at various temperatures [9] a)
3.2. Application to symmetrik top molecuks
‘1 The values of CH2Cl2 are average of results at -80, -40, and 0°C. the range III values are grven in parentheses, ref
PI-
L April 1980
LETTERS
,
and therr ratios px = Dxx/Dzz, py = Dy y/Dzz_ Listed also are the experrmental values for those ratios. Lastly, we have estimated the viscosrty of CS3 at vanous temperatures [ 191 so as to obtam the dtffusion constants for CH$12 in CS;? which are hsted in table 3 wrth v taken to be 6, the experimental values are in parentheses. Inspecting table 3, rt is clear that the Y and 2 components of the diffusion tensor are quite close to the expertmental values. On the other hand, the computed X component overestrmates the friction by roughly a factor of two. In all three components, the temperature dependence of both the theoretical versus experimental values is directly comparable except for the factor of two appeanng in Dxx. Thusindicates that the thermal activation process is directly related to viscosity effects.
.
For the molecules chosen here, viz., CS2, benzene and fluoroform, only the perpendrcular component of the symmetric top drffusion tensor enters. Calculated and experimental results are shown in table 4.
4. Concluding remarks The calculatron presented approximates the rofational diffusion tensor for a few simple systems. The agreement of theory and experiment is encouraging and, frankly, better than expected. Representing a connected array of atoms as hydrodynamically nonTable 4 Onentatronal correlation tnnes for neat
symmetric
top
ffuids~)
Molecule
rl(cP)
T(K)
72Cexe.l
r2kak)
CS2 1111
0 336
273
15
1.05 b)
CHFs
(121
benzene [131
0.425
170
1.26
1.32=)
0 607
296
2.91
3.58 dj
a) A!I molecular parameters have been taken from ref. 1241. b, Determmed using q = 0.78 A, q-~ = 1.554 A. 4 aH = 0 328 A, “C = 0.77 A, q = 0.562 A. ‘CH = L-098 A, rm = 1.332 A, LFCF = 108.5”. d, a;;-= 0.384 A, bC = 0.697 A, Q-H = 1 08 A, ‘cc = 1.39 x. LCCC = 12OO.
Volume 71, number 1
CHEMKAL
mteracting spheres IS an approximatron to a dynamrcal picture of Hu and Zwanzig [20] wherein one solves the linearized Navier-Stokes equations for a nonspherical body subject to slip boundary condrtions. Included in our theory is the room for parameter modrfication. There are two parameters in the friction constant worth further scrutiny. They are the strck(6) or shp(4) coefficient for the translation of the constituent spheres. and, second, the radu of the spheres We chose 6, for stick, and the covalent bond radu for two reasons. First, being hrstonc, the translational diffusion constants for a senes of alkane chams and the drelectnc relaxatton correlatron times for 1-bromoalkanes can be fit by this particular choice Second, rf we rmplement van der Waals radu, the spheres overlap and we must correct our model to account for the exposed part of the surface. On the other band, covalent radu for the most part do not lead to overlappmg of the spheres and hence nught better represent the overall molecular volume. Certaxdy. our so-called construct molecule
is too lumpy, but nevertheless
it is fathful
to
the bond angles and bond lengths. Improvements on the model could be focused along the lure of Happel and Brenner’s method of reflections [17], wherein we begin to account self-consistently for hydrodynamic interactions among the atoms. However, the mathematics rapidly becomes tedious and at tunes physically suspect on account of the need to evaluate the Oseen tensor very close to the particle surface. The overall impact of this note is to stress one pomt. the rotational diffusion tensor can be approxrmately represented by a model of non-interactmg spheres, and the results are surpnsmgly good.
Acknowledgement Partial support of this work by the Public Health Service through grant GM 0852 to DMG is acknowledged. DCK and GTE were supported in part by the National Science Foundation.
Appendix
1 A@1980
PHYSICS LETTERS
IV + 1 “soft” variables are the same m both sets. That IS, we consrder only a transformatron of the 2N + 2 “hard” variables. We have IVa = [(2N + 2)(2N + 2 + 1)]/2 = (AT+ 1)(2N+ 3) undetermmed elements of T. We require T such that all Nb = (N + 1)(2N + 2) = 2(N+ i)* elements of R hs be zero. Thrs imposes Nb constramts on T and allows us to set HIJ = (Rss)zJ for our purposes. Since m general the vanatron of q wrth the soft varrables wrll not be the same as qs (the determinant of the soft part of G = G,,) we impose the additional constraints that
l/* = (ilg:l*)(aiaQ')g,l'* . (1/g"2)(~/~Q'k for all “soft” variables Qi. Thrs introduces N, = N + 1 addrtronal constraints on T. We have then Afb + N, = IV, constraints and have umquely defined T.
Appendix
2
Consider a body composed of N spheres Joined by resistanceless bars and neglect hydrodynamic mteractrons between the spheres. If the body IS moving relatrve to a surroundmg fluid with velocity U and constramed not to rotate the Stokes fnctronal force on sphere i wrll be Fl = 6n-qal(I. Measured from some pomt 0 fLxed in the body, the torque on sphere I is T, = ro, X Fl. The total torque on the body 1s the sum of these torques The torque produced by translatron rotation couplmg is given by -C - U [ 171 where C is the couplmg tensor. Thus
Hence C is an antrsymmetnc tensor with elements proportronal to sums of the type zr alro,z_ A sufficient condition that the coupling vanish is that thrs sum is zero, thus defining the center of resistance. This conclusion IS in agreement with Condrff and Dahler [ 181, who find that even for loaded spheres “the particle is located at the posrtion of rts center of resistance, rather than at the center of mass”.
1 References
Consider a symmetric linear transformation T, from our original set of 3N + 1 genera!ized variables (N the number of bonds) Q’ to a new set q’ such that the 162
[ 11 J-P- Ryckaert (1975) 123:
and A. Bellemans, Chem. Phys Letters 30
Volume 7 1, number 1
CHEMICAL.
I P. Ryckaert, Ph D. Theses, Free Uruversrty of Brussets (1976) [2] M. FLyman. J. Chem Phys 69 (1978) 1527.1538. [3] E Helfand. J. Chem. Phys. 54 (1971) 4651;69 (1978) 1010; E. Heffand, A R. Wasserman and T k Weber, J. Chem.
Phys. 70 (1979) 2016; E. Helfand and J. Skobuck, Kmetrcs of Conformatronal Changes m Polymers, to be pubhshed [4] J H. Wemer and M-R. Pear, Macromolecules 10 (1977) 317, M.R Pear and J.H Werner, J Chem Phys. 71 (1979) 212. [S] M. Brshop, M H. Kalos and H L Fnsch, J. Chem. Phys 70 (1979) 1299, D. Ceperley, M H. Kales and J L Lebow-ttz, Phys Rev.
Letters 41 (1978) 313 [61 R ht. Levy, M Karplus and J.A McCammon. Chem. Phys Letters 65 (1979) 4. [71 G T Evans, Mol Phys. 36 (1978) 1199, J. Chem Phps 69 (1978) 3363, G-T Evans and DC. Knauss. J. Chem. Phys. 71 (1979) 2255,72 (1 Feb 1980). to be pubhshed [8] E Paul and R.M hfazo. J. Chem. Phys. 48 (1968) 1405. [9] D M Grant, MT Chenon. C L Mayne and LG. Werbelow, Proceedmgs of the XX Colloque Spectroscopium Internattonale and 7th Internattonal Conference on Atomic Spectroscopy, Vol. 2, Prague, Czechoslovakia (Sept 1977) p 207.
PHYSICS LETTERS
1 April 1980
[IO]
CL- Mayne, D.M. Grant and D-W. Alderman, I. Chem Phys. 65 (1976) 1684. [ 1l] H.W. Speiss. D. Schweitzer. U. Haeberlen and K-H. Hausscr, J. Magn. Reson. 5 (1971) 101. [12] J. DeZwaan, D-W. Hess and C.S. Johnson, J. Chem Phys 63 (1975) 422. 113J G-R. Alms, D R. Bauer. J-1. Brauman and R. Pecora, J. Chem. Phys. 58 (1973) 5570. [ 141 hf. Fixman and J. Kovac, I. Chem. Phys. 61(1974) 4939. [15] DC. Knauss andG.T. Evans, J. Chem Phyr 72 (I Feb.
1980). to be pubtished. [16] J-G. Kirkwood. J. Polym Sci. 12 (1954) 1. [ 17) 1. Happel and H. Brenner, Low Reynolds number hydrodynamrcs (Prentice-Hall. Englewood Cliffs. 1965) ch. 5. [18j D.W. Condrff and J.S. Dahler, J. Chem. Phyr a (X966) 3988. [I91 lnternationat CritrcaJ Tables. VoL 7 (1930) p. 213. [20] C.-hf. Hu and R. Zwanzig, J. Chem Phys. 60 (1974) 4354. [2li RJ Myers and W.D. Gwinn, J. Chem Phys 20 (1952) 1420. [22] D. Chadwick and DJ. Mrller, Traus. Faraday Sot 67 (1971) 1539. [23] 0. Bastransen, T&&r_ Kjemi, Bergves. nfetafL 6 (1946)
[ 241 kbles of interatonuc drstances and confiiuration in molecules and ions (Chemrcal Society, London, 1958).
163
ROTATIONAL Donald
CHEhIlCAL
I
DIFFUSION
C. KNAUSS,
Department
TENSORS
PHYSICS
LETTERS
FOR NONSPHERICAL
1 Aprd 1980
MOLECULES
Glenn T. EVANS
of Chemurry. Oregon State Unwemty,
Con allis.
Oregon 9 733 I, USA
and Davxd M. GRANT Department
of Che.nirtry.
Received 5 December
1979,
Unhersqv
in foal
of Utah, Salt Luke City, Utah 84i1.2, USA form 15 January
1980
Tbc rotatlonal tifuslon tensors of cweral ngld molecules are determmed from the rotational fnctlon tensor. The molecular friction tensor IS modeled as a sum over the translational Stokes law fnctton coefficients of the constituent atoms of the molecule. Agreement of the theory w~tb expenmental rotatIonal tifusion constants and orlentattonal correlatton times Is surpramgly good.
1. Introduction Browman dynamxs simulations [l-7] of chain molecules have been apphed recently to the calculation of various transport coefficients of flevlble molecules m hquid solvents. At the crux of the brownian motion procedure is a friction constant for the atoms of the parent chain molecule. A pnon It IS not clear how to represent the fricuonal charactenstlcs of constituent atoms in a molecule. Mazo [8] has found that the translational diffusion coefficient of an alkane cham can be quantitatively determined by means of the Kirkwood formula parameterized m terms of the fnction coefficient per backbone atom. The actual fnction coefficient of the backbone atom represents the carbon atom by its covalent ra&us (0.77 A) and stick boundary conditions in the context of Stokes law. Evans and Knauss [7] have found that the dlelectnc correlation firne of a series of I-bromoalkanes can also be accommodated by the same parametenzatlon. With this in mind, it seemed mtngumg to apply the same prescnption to small rigd molecules, such as methylene habdes, CS2, and benzene, since the rotatlonal Mfusion tensors and orientational correlation tunes have been determined [9-l 3] _ The chosen model IS as follows: a rigd molecule is 158
represented by a collectlon of hydrodynamically noninteractmg spheres, fiuted at distances and angles consistent ~th the known molecular structure. The friction coefficient of each atom 1s gven by the translational Stokes law and the overall friction tensor IS then constructed from the appropnate tensor sums of fnctron constants. The molecular diffusion tensor 1s the mverse of the molecular friction tensor. In the following section, we present the theory underlymg this work and an application to a few simple rigid molecules.
2. Theory The specific distnbution function for a tagged molecule in a fluid shall be defined as 9(f), which is a function of all the coordmates of the atoms that constitute the molecule. \k(t) obeys a continuity equahon in Remann space [2,14-161
a,*(t)
= ~l/g1/2)(a/aQ’)[g112hi~(t)]
_
Q’ IS a generalized coordmate, al(r) the dynamical generabzed velocity, and g the covariant metric determinant. The covariant metnc tensor, G,, is
Volume 7 1, number 1
CHEMICAL PHYSICS LE-F-FBRS
3N
(2) where (3)
_
?he Set of d xk, {xk,
k = 1, 3N),
Comprk
the
x,
H = RSS_ Rsh . (Bhh)-1
-
In actual fact, the Qi shall be chosen to be the Euler angles, bond angles, and bond lengths of the molecule. To recast eq. (1) into a drffusion equation, we make use of the force balance equation,
,
(4)
with Fk = -_(a/axk)(kBT
h \k+ (/) .
(5)
Eqs. (4) and (5) relate the total force to the sum of the entropic and intramolecular (or torsional) forces. As wntten in eq. (4), the friction coefficient vector IS 3ZVdimensional and, in principle, one could allow for anisotropic translational friction of each of the atoms of the molecule. This wrll not be done, but rather
{‘Yk,k=1,3N3=Cy,,~=Yn,To,llb,~~,Yg,-.-3Next,
transform
the cartesian
velocity
(ala@ + B aujag+tqt)
to Riemann
space, then
. Bhs ,
(12)
of the “soft-soft” part of the covariant metric C. By defining moments of resistance in a mmner analogous to the moment of inertia (replacing mass by friction constant), we find that the elements of the symmetric covariant tensor S (the inverse of H) are simply related to these moments and the Euler angles. We pick a body-fuced coordinate system which diagonalizes the moment of resistance tensor (i.e., friction tensor) and fiid that the elements of S are [IS1
SOS= K y y sin2a + Kxx cos2cl,
+ K y ,r cos2a sin2j3+ Kzz
A[k = a&axk
(7)
and (8)
(9)
cos2j3 ,
Spr=(KXX-Kyy)cos~sinasinB, Sa7=KzzcosP,
k,T ln aI) _
As a result, the velocity in generalized coordmates is
and
(IV
s Y-Y = Kxx sin’s sin20
where
-(a/a&(u+
P= W,T,
RhS being the “hard-soft” part of the tensor R. Thus the implied sums in eq. (11) go over only the “‘soft” variables. For a heterogeneous fle_xible chain molecule, the soft vanables would consist of the thee Euler angles and the torsion angles (assuming that bond lengths and nearest neighbor bond angles are fxed)For rigid systems, the soft variables are only the tf.1122 Euler angles, and Q is just the distribution function of the Euler angles. We can show (see appendix 1) that the set of “hard” variables (combinations OF the bond lengths and bond angles) can be chosen such that Rhs = 0 and g = gs, where gs is the determinant
s P(L =Kzz,
fi =
,
where H is the contravariant tensor,
molecule. Thus, the xk denote the ordered set, Cq, ‘2, --- rrl
x
y,
and z Cartesian coordinates of the N vectors, r,, that specrfy the location of the N atoms comprisiig the
= +Fk
If we now implement constraints on the so-called “hard” variables in the usual manner [2,14-161, we find
a,@(t) = (kT/g’12)(W3Qi)g’12H~
Blk = axkjaQ’
&
1 ApilL
Sap=%
03)
with
H=S-’
_
04)
Q, p, and 7 are the Euler angles and are defined in accord wi*& our previous work [IS] _The K tensor is akin to an inertia tensor with the friction constant playing the role of mass; hence,
159
1 Aprit 1980
CHEMICAL PHYSICS LETTERS
Volume 71, number I N
K=C y (AL-rJrJ) J J
,cl
in the above context, we are associating an tsotroptc transiattonal frrctton constant wrth each of the con-
stituent N atoms. rather than wrth each of the 3X coordmates. Folfowmg earher work [ 151, one tnverts S to form a diffusron equation 2,8(f)
= r.I * D
l
+5’(t)
,
WI
IJ is the lab-fLsled rota&on operator rotatronal diffusron tensor, D,J =k,T
$J&,
and D is the
-
which IS given ehphcitly rn terms of the elements of the friction tensor, and consequently the frrctron constants of the constituent partrcles. Eq (15) doffers from Kirkwood’s diffuston tensor ordy rn as much as we have allowed the constrtuent atoms to have ddferent frictron constants.
3. Comparison
with experiment
3.1. Appkatron
to nzethylerre halides / CH2Xz)
To apply eq. (16), we must dtagonahze thi friction tensor. K. Consequently, define our body-fixed coordmate system for methy lene hahdes as follows- The 2 axis is parallel to the lute Jouung the two halogens, the Y aus IS parallel to the hne ~ommg the two hydrogcns. and the X axis is perpendrcular to both. Referrrng to fig_ 1. we see that the various atoms have pontion vectors
0. A),
rxt=(B-D, r&=(B+F, rc =(B,
-.t?, o),
rxz=(B-D, ‘H~=(B+F.
for the translational drag on the kth atom, yk = mLrkn% where v = 6(4) for stick(shp) boundary condrtrons and ak is the appropnate radius of the kth particle. then the elements of the fnction tensor become Kxxfvml
= 2(aHEr +aXA2),
K,,/znrq
= 2[aXA2
+aX(D
-B)’
+a~(B+l=)~]
w?,B2,
Kzzjvnq
= 2[aHE2 + aH(B + F)* + ax(D - B)2j
+a,B’,
Table 1 Geometnc parameters (angles in degxe, distances m A) used m the methytene halide cakulatlons
E, o),
0, 0).
X
Fag 1. Coordmate system for the methylene habdes.
0. -A).
Thus defined. the fnctron tensor is dragonal, mdependent of the choice of B. B IS chosen wrthout ambiguity as a uruque point m the body. vrnz.,tire center of resistance, which has the property of decoupling translatmnal and rotational motions (see appendtx 2) m the drffusroaal regrme It is noteworthy that the center of mass of the molecule does not enter the descrrption of the brownian dynanucs, but rather rts center of resistance [ 17,I 81. fmplementing the Stokes relation 160
“‘1
(16)
LX-C-X f. H-C-H IC-Xl E-HI
ax “C OH
C&$32
CHzBr2
(32
1118b) 106.9 1 772 b, 1.079 1.002 0.770 0 310
112 7 =) 105 9 1.927 =) 1.079 1.157 0.770 0 310
114.7 d) 103.6 2.12 d) 1.079 1.350 0 770 0 310
12
a) ar_-taken to be the covalent radms of carbon. OX, aH found from UC and the bond lengths The angle LH-C-H estxmated from the reauuement that tX-C-H retam the tetrahedral value b, Ref (211 c, Ref [221. d, Ref. [23].
Volume 71, number 1
CHEMICAL
PHYSICS
Table 2 Reduced dtffusion constants for methylene hahdes CHsC12
CHaBra
CH2 I2
Dirx
0 209
0 156
0111
D’YY
0.172
0.130
0 094
Diz
0 512
0 453
0 399
PX
0.408
PX(exp ) PY P y(exp.)
0 344
0 180 a) (0 166 - 0.199) 0.336 0 250 a) (0.242 - 0 258)
0 278 0205-~0003~)
0.287
0.236
-
0079=0057b)
b, Ref
[IO]
n (cP)
- a#)/[2(ax
+ aH) + ac]
-
The remaming parameters are readdy determined gven the bond angles and bond lengths. The parameters used are listed in table 1 to obtam the results of table 2, namely, the reduced drffusron constants
D:, = W-v/kTW,,
0°C
-4OT
-80°C
0.43
0.62
0.99
Dxx
X lOto s-l
9.7 (5.4)
5.7 (2.9)
3.0 (1.2)
Dyy
X 10”
8.1 (6.6)
4.8 (4.3)
2.5 (L-9)
Dzz
X 1O’O s-r
24 (27)
14 (17)
7-4 (7.5)
s-r
a) Expertmental values rn parentheses.
The second rank orientational correlation time, ‘2, is related to D, in the brownian limit by 72 = l/60,
where B = 2(f7,D
Table 3 Calculated and experimental diffusion constants for CDaCLa at various temperatures [9] a)
3.2. Application to symmetrik top molecuks
‘1 The values of CH2Cl2 are average of results at -80, -40, and 0°C. the range III values are grven in parentheses, ref
PI-
L April 1980
LETTERS
,
and therr ratios px = Dxx/Dzz, py = Dy y/Dzz_ Listed also are the experrmental values for those ratios. Lastly, we have estimated the viscosrty of CS3 at vanous temperatures [ 191 so as to obtam the dtffusion constants for CH$12 in CS;? which are hsted in table 3 wrth v taken to be 6, the experimental values are in parentheses. Inspecting table 3, rt is clear that the Y and 2 components of the diffusion tensor are quite close to the expertmental values. On the other hand, the computed X component overestrmates the friction by roughly a factor of two. In all three components, the temperature dependence of both the theoretical versus experimental values is directly comparable except for the factor of two appeanng in Dxx. Thusindicates that the thermal activation process is directly related to viscosity effects.
.
For the molecules chosen here, viz., CS2, benzene and fluoroform, only the perpendrcular component of the symmetric top drffusion tensor enters. Calculated and experimental results are shown in table 4.
4. Concluding remarks The calculatron presented approximates the rofational diffusion tensor for a few simple systems. The agreement of theory and experiment is encouraging and, frankly, better than expected. Representing a connected array of atoms as hydrodynamically nonTable 4 Onentatronal correlation tnnes for neat
symmetric
top
ffuids~)
Molecule
rl(cP)
T(K)
72Cexe.l
r2kak)
CS2 1111
0 336
273
15
1.05 b)
CHFs
(121
benzene [131
0.425
170
1.26
1.32=)
0 607
296
2.91
3.58 dj
a) A!I molecular parameters have been taken from ref. 1241. b, Determmed using q = 0.78 A, q-~ = 1.554 A. 4 aH = 0 328 A, “C = 0.77 A, q = 0.562 A. ‘CH = L-098 A, rm = 1.332 A, LFCF = 108.5”. d, a;;-= 0.384 A, bC = 0.697 A, Q-H = 1 08 A, ‘cc = 1.39 x. LCCC = 12OO.
Volume 71, number 1
CHEMKAL
mteracting spheres IS an approximatron to a dynamrcal picture of Hu and Zwanzig [20] wherein one solves the linearized Navier-Stokes equations for a nonspherical body subject to slip boundary condrtions. Included in our theory is the room for parameter modrfication. There are two parameters in the friction constant worth further scrutiny. They are the strck(6) or shp(4) coefficient for the translation of the constituent spheres. and, second, the radu of the spheres We chose 6, for stick, and the covalent bond radu for two reasons. First, being hrstonc, the translational diffusion constants for a senes of alkane chams and the drelectnc relaxatton correlatron times for 1-bromoalkanes can be fit by this particular choice Second, rf we rmplement van der Waals radu, the spheres overlap and we must correct our model to account for the exposed part of the surface. On the other band, covalent radu for the most part do not lead to overlappmg of the spheres and hence nught better represent the overall molecular volume. Certaxdy. our so-called construct molecule
is too lumpy, but nevertheless
it is fathful
to
the bond angles and bond lengths. Improvements on the model could be focused along the lure of Happel and Brenner’s method of reflections [17], wherein we begin to account self-consistently for hydrodynamic interactions among the atoms. However, the mathematics rapidly becomes tedious and at tunes physically suspect on account of the need to evaluate the Oseen tensor very close to the particle surface. The overall impact of this note is to stress one pomt. the rotational diffusion tensor can be approxrmately represented by a model of non-interactmg spheres, and the results are surpnsmgly good.
Acknowledgement Partial support of this work by the Public Health Service through grant GM 0852 to DMG is acknowledged. DCK and GTE were supported in part by the National Science Foundation.
Appendix
1 A@1980
PHYSICS LETTERS
IV + 1 “soft” variables are the same m both sets. That IS, we consrder only a transformatron of the 2N + 2 “hard” variables. We have IVa = [(2N + 2)(2N + 2 + 1)]/2 = (AT+ 1)(2N+ 3) undetermmed elements of T. We require T such that all Nb = (N + 1)(2N + 2) = 2(N+ i)* elements of R hs be zero. Thrs imposes Nb constramts on T and allows us to set HIJ = (Rss)zJ for our purposes. Since m general the vanatron of q wrth the soft varrables wrll not be the same as qs (the determinant of the soft part of G = G,,) we impose the additional constraints that
l/* = (ilg:l*)(aiaQ')g,l'* . (1/g"2)(~/~Q'k for all “soft” variables Qi. Thrs introduces N, = N + 1 addrtronal constraints on T. We have then Afb + N, = IV, constraints and have umquely defined T.
Appendix
2
Consider a body composed of N spheres Joined by resistanceless bars and neglect hydrodynamic mteractrons between the spheres. If the body IS moving relatrve to a surroundmg fluid with velocity U and constramed not to rotate the Stokes fnctronal force on sphere i wrll be Fl = 6n-qal(I. Measured from some pomt 0 fLxed in the body, the torque on sphere I is T, = ro, X Fl. The total torque on the body 1s the sum of these torques The torque produced by translatron rotation couplmg is given by -C - U [ 171 where C is the couplmg tensor. Thus
Hence C is an antrsymmetnc tensor with elements proportronal to sums of the type zr alro,z_ A sufficient condition that the coupling vanish is that thrs sum is zero, thus defining the center of resistance. This conclusion IS in agreement with Condrff and Dahler [ 181, who find that even for loaded spheres “the particle is located at the posrtion of rts center of resistance, rather than at the center of mass”.
1 References
Consider a symmetric linear transformation T, from our original set of 3N + 1 genera!ized variables (N the number of bonds) Q’ to a new set q’ such that the 162
[ 11 J-P- Ryckaert (1975) 123:
and A. Bellemans, Chem. Phys Letters 30
Volume 7 1, number 1
CHEMICAL.
I P. Ryckaert, Ph D. Theses, Free Uruversrty of Brussets (1976) [2] M. FLyman. J. Chem Phys 69 (1978) 1527.1538. [3] E Helfand. J. Chem. Phys. 54 (1971) 4651;69 (1978) 1010; E. Heffand, A R. Wasserman and T k Weber, J. Chem.
Phys. 70 (1979) 2016; E. Helfand and J. Skobuck, Kmetrcs of Conformatronal Changes m Polymers, to be pubhshed [4] J H. Wemer and M-R. Pear, Macromolecules 10 (1977) 317, M.R Pear and J.H Werner, J Chem Phys. 71 (1979) 212. [S] M. Brshop, M H. Kalos and H L Fnsch, J. Chem. Phys 70 (1979) 1299, D. Ceperley, M H. Kales and J L Lebow-ttz, Phys Rev.
Letters 41 (1978) 313 [61 R ht. Levy, M Karplus and J.A McCammon. Chem. Phys Letters 65 (1979) 4. [71 G T Evans, Mol Phys. 36 (1978) 1199, J. Chem Phps 69 (1978) 3363, G-T Evans and DC. Knauss. J. Chem. Phys. 71 (1979) 2255,72 (1 Feb 1980). to be pubhshed [8] E Paul and R.M hfazo. J. Chem. Phys. 48 (1968) 1405. [9] D M Grant, MT Chenon. C L Mayne and LG. Werbelow, Proceedmgs of the XX Colloque Spectroscopium Internattonale and 7th Internattonal Conference on Atomic Spectroscopy, Vol. 2, Prague, Czechoslovakia (Sept 1977) p 207.
PHYSICS LETTERS
1 April 1980
[IO]
CL- Mayne, D.M. Grant and D-W. Alderman, I. Chem Phys. 65 (1976) 1684. [ 1l] H.W. Speiss. D. Schweitzer. U. Haeberlen and K-H. Hausscr, J. Magn. Reson. 5 (1971) 101. [12] J. DeZwaan, D-W. Hess and C.S. Johnson, J. Chem Phys 63 (1975) 422. 113J G-R. Alms, D R. Bauer. J-1. Brauman and R. Pecora, J. Chem. Phys. 58 (1973) 5570. [ 141 hf. Fixman and J. Kovac, I. Chem. Phys. 61(1974) 4939. [15] DC. Knauss andG.T. Evans, J. Chem Phyr 72 (I Feb.
1980). to be pubtished. [16] J-G. Kirkwood. J. Polym Sci. 12 (1954) 1. [ 17) 1. Happel and H. Brenner, Low Reynolds number hydrodynamrcs (Prentice-Hall. Englewood Cliffs. 1965) ch. 5. [18j D.W. Condrff and J.S. Dahler, J. Chem. Phyr a (X966) 3988. [I91 lnternationat CritrcaJ Tables. VoL 7 (1930) p. 213. [20] C.-hf. Hu and R. Zwanzig, J. Chem Phys. 60 (1974) 4354. [2li RJ Myers and W.D. Gwinn, J. Chem Phys 20 (1952) 1420. [22] D. Chadwick and DJ. Mrller, Traus. Faraday Sot 67 (1971) 1539. [23] 0. Bastransen, T&&r_ Kjemi, Bergves. nfetafL 6 (1946)
[ 241 kbles of interatonuc drstances and confiiuration in molecules and ions (Chemrcal Society, London, 1958).
163