Microscale Molecular Tennis'
L.J.F. Hermans
Huygens Laboratory, Leiden University, P.O. Box 9504, 2300RA Leiden, The Netherlands
Abstract. Most of the cutting-edge research on molecule-surface interactions has been taken over by molecular-beam experiments. Even so, work on transport properties—much in the spirit of Lloyd Thomas—has been yielding subtle information
on the role of the internal state that is difficult to obtain by other methods. Two such transport experiments are highlighted.
One is the influence of an external magnetic field on a Knudsen flow. It gives information on nonequilibrium angular momentum polarizations produced by nonspherical molecule-surface interactions. The other is Surface Light-Induced Drift. It gives
information on the rotational and vibrational-state dependence of accommodation coefficients. This includes the role of the
direction of the angular momentum with respect to the surface ("helicopter" vs. "cartwheel").
INTRODUCTION
Our earliest understanding of the behavior of molecules at solid surfaces was derived largely from experiments on
transport properties. The pioneering work of Lloyd B. Thomas, R.G. Lord and many others has yielded a wealth of
data on energy and momentum accommodation since the 1940s [1,2]. Much of this work has been reviewed by H.Y.
Wackman [3]. A drawback of this type of experiments is the difficult control over surface quality, since the data are
obtained under non-UHV conditions by definition.
The introduction of molecular-beam techniques in combination with Ultra High Vacuum equipment opened up
possibilities to study molecule-surface interactions under well-defined surface conditions. In addition, beam data are
much more detailed since they employ selected velocities rather than a near-Maxwellian. The advent of narrow-band
tunable lasers has added state-specificity to many beam data; one may say that they provide information on scattering
kernels rather than on the scattering operator. A consequence is that a direct comparison between beam data and
transport data is no longer possible: molecular beams produce so-called "beam accommodation coefficients" rather
than the traditional "global" accommodation coefficients, which are obtainable only by integration over different
directions and energies. It is fortunate that the "unified kinetic approach" developed by Borman, Krylov and coworkers
does build a bridge between the two types of data sources (see, e.g., [4, 5, 6]).
Although much of the cutting-edge research on molecule-surface interactions has been taken over by molecularbeam experiments, two transport-based experimental techniques were developed over the last two decades which yield
rather unique information on the role of the internal state on the scattering process. One is the influence of an external
field on Knudsen transport properties. It employs precession of the molecules in an external field. The other is Surface
Light-Induced Drift. It employs velocity-selective excitation of molecules by a narrow-band tunable IR-laser. It is
the purpose of this paper to briefly discuss both types of experiments, and to highlight the information that they have
provided.
MAGNETC FIELD EFFECT ON KNUDSEN FLOW
In any Knudsen transport situation, the nonequilibrium velocity distribution will, by nonspherical molecule-surface
interaction, also cause the angular momentum distribution to be anisotropic. This coupling between v -space and Jspace will have its impact on the transport. This can be made visibe by changing the anisotropy in J-space using the
1
Lloyd Thomas Memorial Lecture
CP663, Rarefied Gas Dynamics: 23rd International Symposium, edited by A. D. Ketsdever and E. P. Muntz
© 2003 American Institute of Physics 0-7354-0124-1/03/$20.00
10
(a)
Without field____________
(b) With field
FIGURE 1. Elementary picture of the field effect on a Knudsen flow, using the analogy of a ball bouncing from a rough surface,
(a) The spin produced by tangential forces during the first collision yields a drawback for the forward motion in the second, (b)
Precession of the rotational axis between two collisions, caused by an external magnetic field, removes the drawback. For the gas
as a whole this enhances the flow.
precession of the molecules around an external field: this will, in turn, change the velocity distribution, and thus the
transport.
The theory of such effects was worked out by Borman et al. [7], by Knaap and Kuscer [8] and by Cercignani [9].
The field effect on a heat flux in a Knudsen gas was first observed by Borman et al. [10]. Here we will discuss the
magnetic field effect on a Knudsen particle flow through a flat channel.
An elementary picture of the effect is given in Fig. 1. To achieve precession of the molecules around the field, use is
made of the (small) magnetic moment resulting from the rotation of the molecule: JJL = g j-fiN, where g is the molecular
Lande-factor and p,jy the nuclear magneton.
In an external field, this produces a (J-independent) precession frequency u = (gp>N/h)B, with B the field strength.
The relevant parameter for the field dependence is the precession angle CJT, where r is a typical flight time between
the two parallel surfaces forming the channel: r — b/(2kT/m)1^2 with b the spacing between the surfaces.
The field dependence of the change in the Knudsen flow, £$/<!>, would have a simple (1 — COSCJT) behavior if r
would be uniform throughout the gas. Taking the Maxwell velocity distribution into account, one expects a behavior
of the form
6$
f°°
— ~F(LJT) = I [(1 — COSUJT)/V]Vexp(—v )dv
(1)
^
Jo
where v represents the dimensionless z-velocity of a molecule, with z along the surface normal. However, for a
complete description of the effect, one has to take into account that various types of J-polarization can be produced,
which can give different contributions to <5<!>/<I>.
It should be stressed at this point that, according to the Curie principle, a particle flow, having a vector character,
can generate only polarizations (in the combined velocity-angular-momentum space) which are true vectors. In
the kinetic description of ref. [8], only the four simplest polarizations were considered, viz., two of first rank in J
("orientation") and two of second rank ("alignment"). These are listed in Table 1. Note that orientation and alignment
give contributions with opposite signs, and that alignment—due to its symmetry—can produce a "double-frequency
behavior", i.e., F(GJT).
The experiments were performed in a capillary Wheatstone bridge arrangement, with one of the capillaries replaced
by a flat channel in a magnetic field. The field effect is then detected as an off- balance, and measured in terms of a
small pressure difference. For a description of the experiment see, e.g., [11,12, 13] and references therein.
The role of the molecular symmetry
Examples of experimental results are shown in Fig. 2 for two different molecules and the same LiF(OOl) surface.
It is seen that the two gases have markedly different behavior: CH4 behaves almost perfectly according to the Jy
polarization, i.e., like the elementary tennisball picture in Fig. 1. This seems to reflect the "rough-sphere"-like behavior
of CH4. By contrast, for H2 we see (both from the sign of the effect and the position along the c<;r-axis) a clear
dominance of alignment types of polarization, viz., the lower two in Table 1. This seems to reflect the "dumbbell'Mike
behavior of diatomic molecules, since these alignment types require torques along the z-axis during the collision,
which are absent, by symmetry, for a spherical molecule.
11
TABLE 1. Four types of polarization which can be produced
in a Knudsen flow (along z-axis) between parallel plates (surface normal n along z). The behavior of their contributions to
the field effect 63>/$ is given for three field orientations (cf.
Eq. (1)). Note that the polarizations are actually x-components
of true vectors (J itself is a pseudovector). Example: Jy =
(J x n) x , which is the elementary "tennis ball" type shown in
Fig. 1.________________________________
Type of polarization_____B||rr___
vyJz
F(UJT)
-F(2vT)
JxJz
-J
0.2
(a)
CH4/LiF(001)
0.1
O
-0.1
10
0.1
U)T
FIGURE 2. The change in flow £<$/$ measured as a function of precession angle ur for CHi (field along z) and H2 (field along
y). The surface is LiF(OOl) with the flow along the [100] direction. The curves through the data points are theoretical curves (see
Eq. (1) and Table 1) with the vertical scale as the only free parameter.
The role of surface corrugation
Tangential forces acting during the molecule-surface collision are obviously essential for this phenomenon: no
such effects can arise for a perfectly smooth surface. The question then arises what role surface corrugation plays. To
investigate this, experiments were done using a LiF(OOl) surface with two orientations with respect to the crystal axes.
This makes it possible to vary the corrugation for essentially the same surface. Details of the set-up are given in [13].
Data obtained from this experiment are shown in Fig. 3 for H 2 , HD and N2. When comparing the left-hand panel with
the righ-hand panel for each gas, it is seen that the role of corrugation is remarkably large for H2, but much smaller
forHDandN 2 .
The explanation of this behavior may be given in the framework of the unified theory discussed in Refs. [4] and
[5]. For H2 the large rotational level splitting (H2/2Ik = 85 K) combined with the selection rule A J = 0, ±2 requires
a large energy exchange with the phonon bath if the rotational state is to be changed. The interaction range effective
for this process is limited to the small region of closest approach, where the interaction is strong. It is in this part of
the interaction range that the corrugation makes itself felt. Thus, an appreciable influence of the corrugation is to be
expected. For a molecule like HD, however, the smaller level spacing and the possibility of A J = 1 collisions makes
rotational transitions occur over a much larger region of the trajectory. Therefore the molecules, while receding from
the surface, can continue to interact with the surface up to distances where the influence of the corrugation is effectively
washed out. This argument is corroborated by the fact that H2—although rather spherical in its interaction—showed
larger field effects than did HD or N2 : it "benefits" more from the corrugation than do heavier molecules having
12
i i mi———i—i—r
i i i ni———i—i—r
H2/LiF(001)
Flow/[100]
H2/LiF(OOl)
Flow/[110]
1CT
1.5
I
HD/LiF(001)
Flow/[100]
1.0
I I I ll|————I——I—I
I I III]—————I——I—!
I I I I l|
HD/LiF(001)
F!ow/[110]
0.5
^P
-0.5
-1.0
'
I I Itit
———I———t
I I I I MI
10"
3
do- )
1
I
CJJT
l i f t .
10-
10
N2/LiF(001)
Flow/[100]
10
N2/LiF(001)
Flow/[110]
U)t
10
FIGURE 3. Data for H2, HD and N2 flowing along a LiF(OOl) surface, with two crystal orientations (see inset). Each panel
shows data for two field orientations. Left-hand panels: flow along [100] (higher corrugation). Right-hand panels: flow along [110]
(lower corrugation).
smaller level spacing.
SURFACE LIGHT-INDUCED DRIFT
Consider a Knudsen gas at rest in a tube which is sealed off by optical windows. Let us shine a narrow-band tunable
laser down the tube, tuned into the wing of the Doppler-broadened absorption profile of some ro-vibrational transition.
This will produce velocity-selective excition (see Fig. 4). If the tangential momentum accommodation coefficient is
state dependent, a net tangential momentum transfer to the surface will take place, and the gas will be set into motion.
In a closed tube this will give rise to a pressure difference, which can be related to the difference in accommodation
coefficient (see e.g., Ref. [14] for details).
Even if ro-vibrational excitation involves a change in both vibrational and rotational quantum number, it has
13
vL(=Aco/k)
excited state
v
k
ground state
Laser
pressure difference
FIGURE 4. The principle of Surface Light Induced Drift. Shown is the case of laser detuning in the blue wing which excites
molecules moving to the right. Here, excited molecules are assumed to have larger accommodation.
x
a
<
-1
-2
DR(J-1)
OP(J)
* average
-3
-4
FIGURE 5. The change in accommodation coefficient for tangential momentum upon excitation, Aa, as a mention of rotational
quantum number J. Results are shown for P( J) and R( J - 1) transitions in the fundamental vibrational band v — 0 -> 1. Note that,
for each J, the two data points correspond to opposite J transitions.
been shown [15] that the two contributions can often be separated into a rotational and a vibrational part: Aa =
Aa roi + Aayi&. This is achieved by combining the results for P( J) and R( J - 1) transitions. For such pairs, the
changes in J are equal but opposite, while the change in v (v = 0 -)• 1) is the same. An example for HF interacting
with a stearic acid coated surface is shown in Fig. 5. For each pair of data points, the average can be interrpreted as
Aa v i6, while half the difference corresponds to Aa roi .
From the data it is found that a decreases slightly upon vibrational excitation (by almost 1 x 10~3 in the above
case). This may be qualitatively explained by vibrational-to-translational energy transfer arguments [15]. By contrast,
a is found to increase with increasing J for the case studied. A qualitative explanation along the lines of the unified
approach [4] is under way [5, 6, 16].
14
0.4
;OCS:P(5)
n e /n
0
1 II I I II I I
-0.2
10'
P (Pa)
10*
FIGURE 6. Results of SLID experiments in a plane parallel geometry for OCS interacting with a glass surface. Shown is the
normalized observed pressrue difference for ro-vibrational excitation ^ = 0 — » 2 , J = 5-»4. Black circles denote laser polarization
parallel to the surface, open squares perpendicular. The difference between the data points corresponds to "helicopters" having the
lower accommodation.
Helicopters vs. Cartwheels
We will now address the question how the accommodation coefficient depends on the orientation of the rotational
axis with respect to the surface. This can sensitively be studied by using a flat channel in combination with the
excitation by linearly polarized light. Details of this experiment can be found in [17]. Results for the linear molecule
OCS interaction with a glass surface are shown in Fig. 6.
It is seen that in the data points with laser polarization perpendicular to the surface are consistently higher than
those with the light polarization parallel. This corresponds to lower accommodation for the molecules having their
rotational axis perpendicular to the surface ("helicopters"). This result may seem at odds with molecular-beam data
for NO interacting with Pt( 111), obtained by Jacobs et al. [ 18, 19]. They found that NO molecules desorb preferentially
as "helicopters" for high J- values (J > 30). Given the relation between various accommodation trapping coefficients
(as different moments of the same relaxation probability function [4]) one may conclude that NO would also have
higher accommodation for tangential momentum in the "helicopter" mode.
An explanation may be given by considering the role of rotational (de)excitation in the collision dynamics. We will
illustrate this for the model of a linear "dumbbell"-like molecule approaching a smooth surface. Let us first consider an
incoming molecule having a low J (J « J thermal)- This molecule will tend to suffer rotational excitation, thereby
losing part of its "normal energy". The lower value of the outgoing velocityu vz will increase the effective flight
time t and thus also a. However, this mechanism fails to increase a for molecules in the helicopter mode: since
these have their internuclear axes always parallel to the surface, the absence of a torque will leave the rotational state
of the molecule unaffected (model calculations elucidating this point may be found, e.g., in Refs. [20, 21]. Thus,
for low-J molecules, this mechanism gives the cartwheel mode a higher a than the helicopter mode. Let us now
consider incoming molecules with J » J thermal- Here, for molecules in the cartwheel mode, the opposite will
take place: rotational de- excitation will tend to increase the outgoing "normal energy" and thus decrease a. Again,
incoming molecules in the helicopter mode will be unaffected. Thus, for high-J molecules, this mechanism will give
the helicopter mode the higher a, in agreement with experiment.
ACKNOWLEDGMENTS
The author pays tribute to the memory of Jan J.M. Beenakker and Ivan Kuscer, who were closely involved in this field
of research. Furthermore he is indebted to Sergey Yu. Krylov and Anneke Aschoff for many stimulating discussions,
and to the graduate students who carried this research: J.J.G.M. van der Tol, R. Home, R.W.M. Hoogeveen, G.J. van
der Meer, B. Broers and EJ. van Duijn.
15
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