In all cases of interest to us here we are dealing with a problem whose geometry is essentially one-dimensional - an adsorbed film sandwiched be- tween a semi-infinite gas and a substrate as sketched in fig. 5. Desorption proceeds when heat is injected into the film from the substrate, which may be either a metallic heater or crystal surface. The heated film evolves in time toward a steady state at an elevated temperature and reduced thickness. The object of the model is to predict that evolution based upon the simplest possible assumptions.
The basic strategy is to treat the adsorbed film, wherever possible, as a thin slab of bulk liquid, in instantaneous internal thermal equilibrium, and having known properties. For example, in equilibrium at temperature T0 a film of thickness 80 is assumed to have chemical potential
(I) where µ.1(T0 ) is the chemical potential of liquid 4He at T0 and y/8J is the attractive Van der Waals potential due to the substrate. It follows that the
0
0
Jo O
J, H-alJ,
l ! t
0--;-- --- -1--- -- T
.£
1
SUBSTRAT,E,
Fig. 5. Basic features of the model. Adsorbed film is treated as a thin slice of bulk liquid in equilibrium at instantaneous temperature 1( and thickness Br. Energy flux, dU/dt, enters the film from the substrate. Mass and kinetic energy leave the film crossing a reference boundary into the gas, which remains at ambient conditions T0 and Pg. The fluxes J are calculated from simple kinetic theory treating the gas as ideal.
D. Goodstein, M. Weimer/ Simple model of helium desorption kinetics
pressure in equilibrium with the film is
Pg= Po( To) e-y/o6kaTo, (2)
where P0 ( T0 ) is the liquid vapor pressure at T0 •
The statement of conservation of energy for the film-gas system as a whole is simply
dU= dUr + dUg, (3)
where U is the energy injected per unit area from the substrate, and subscripts f and g refer to film and gas respectively. Since the film itself is regarded as a subsystem internally in thermodynamic equilibrium at all times, we may write [21]
dUr = 1f dSr + /J.r dNr,
where Sr is the film entropy, and Nr is given by Nr=n8r.
(4) (5) Here n is the number density in the film, assumed constant and given by the bulk liquid density, and Tr and 8r are the instantaneous film temperature and thickness. All extensive quantities are per unit area. Taking Sr= Sr(Tr, Nr) one finds
(6) where C N is the isosteric heat capacity. In the spirit of the model, we have used s1, the specific entropy of liquid helium in place of (oSr/oNr)r,·
The gas, treated as ideal throughout, is thought of as an infinite reservoir whose temperature, T0 , and pressure, Pg, are unaffected by the desorption taking place. Referring again to fig. 5 we assume the mean free path in the gas, .\, to be sufficiently large that we may imagine a plane a distance I above the substrate, subdividing the film and gas systems, such that I « .\, y / 13 « k 8 T0 , and including a negligibly small number of gas atoms. These conditions are easily satisfied for most cases of interest with I - 100
A.
We thus envisage energy exchange between the subsystems as mediated by two independent streams of particles, one from the gas into the film and one from the film into the gas, unaffected by each other and unaffected by the Van der Waals potential exerted by the substrate. The flux of gas atoms crossing the plane from above is then given by1;
=
Pg/(2'1Tmk8T0 )112•These atoms carry energy per unit time
Q;
= Pg (2k 8T0/'1Tm) 112,where mis the mass of a 4He atom.
(7)
(8)
D. Goodstein, M. Weimer / Simple model of helium desorption kinetics
Supposing that all of these atoms are either adsorbed, with constant probability a, or elastically reflected with probability I - a, the outgoing flux across the reference plane is given by
(9) where J0 represents the number of particles evaporated from the film per unit time per unit area. J0, obtained by considering the gas that would be in equilibrium with the film at temperature Tr and thickness 8r, is
( 1/2
J0 = aPr/ 27Tmk8Tr) , ( 10)
with Pr given by p = P. (T.) e-y/6/k0T1
r 0 r ' ( 11)
and where
(12) is the corresponding energy flux. We simplify matters further by adopting the value a= I independent of velocity because experiments indicate that to be very nearly correct for bulk liquid helium [22]. The net energy transported into the gas is then
dU& =
(Q
0 -Q;)
dt.Substituting eqs. (6), (8), (12), and (13) into the statement conservation, eq. (3), yields
dU dTr ( ) d8r
(2ks)
112{ 1;2 1;2)dt=CNdt+ P.r+Trs1 ndt+ 1Tm PrTr -PgTo .
(13) of energy
( 14) Each term in ( 14) has an appealingly direct physical interpretation. The energy injected into the film-gas system (left hand side) is used up in warming the film (first term on the right), in latent heat when there is net desorption (second term) and in the excess kinetic energy of warm desorbing atoms over cool.adsorbing atoms (third term).
Combining eqs. (5), (7), and ( 10), provides a second equation expressing the conservation of mass,
- n d8 /r dt = (21Tmk )-B 112{ Pr.- 1r r 12 - Pr,-g 0 112) • ( 15) The response of the film is governed by these two fundamental equations.
CN and µ1 depend on both Tr and 8r, s 1 is a function of Tr alone, and all quantities with subscript f may change with time. Noting that Pr may be eliminated by means of eq. ( 11) and using
µ1 = µ1(Tr )-y/8(, (16)
the problem can thereby be reduced to a pair of coupled, nonlinear differential
D. Goodstein, M. Weimer/ Simple model of helium desorption kinetics
equations, (14) and (15), which yield the dynamical behavior of Tr and or for any known heat flux, dU /dt.
2.2. The heat flux term
Let us now consider the specific experimental arrangement of figs. 1 and 3.
The helium film is adsorbed on an ohmic heater which is itself a thin metallic film (typically < 103
A)
evaporated onto a crystal substrate. We assume that power dissipated in the heater at a rate Wis radiated either into the film-gas system or into the crystal according toW = a k ( T: - 1(4 ) + ac ( T: - T04 ) , ( 17) where Th is the heater temperature. We also assume that the crystal, like the gas, is an infinite reservoir whose temperature is always T0 . For small tempera- ture differences this becomes
Th - Tr Th - To
W= R + R
k c
( 18) where the coefficients ak and ac, effective Stefan-Boltzmann constants for phonon radiation, are related to the Kapitza resistances Rk and Rc via
( 19) Re or ac which relates to the interface between the heater and the crystal may be dependably computed from acoustic mismatch theory [23]. In the spirit of the model, Rk, between heater and film, is taken as the measured Kapitza resistance between the heater material and liquid helium. We then have
(20) With W a known experimental quantity, Th is a third dynamical variable along with Tr and Br, and (17) another equation supplementing (14) and (15). This allows through (20) the elimination of dU /dt from ( 14). We note eq. ( 17) assumes implicitly that the intrinsic time constant of the heater, Ch R k R c( R k +
Re>-
1, typically of order 10 ns with Ch the metal film heat capacity, is much less than any other time in the problem.Experiments in the phonon reflection geometry of fig. 2 are of an essentially different nature, requiring an alternate form for dU / dt. We imagine a phonon energy flux,
k
(energy / time· area) incident at the upper crystal surface.Taborek and Goodstein [16] have shown that a fraction of this flux, (I -TJ)R, is specularly reflected. We assume the remaining fraction, TJR, to be absorbed at the interface. The absorbed phonons heat the film to temperature Tr, causing energy to be radiated back into the crystal at a rate ak(Tr4 - T04) so that
(2la)
D. Goodsrein, M. Weimer / Simple model of helium desorprion kinerics
Since in experiments of this kind the energy flux is typically very much smaller than with the film adsorbed directly on the heater, a linear approximation seems likely to be justified. Under these circumstances we obtain
dU/ dt = 71R - (1(- T0)/ Rk to replace dU/dt in eq. (14).
2.3. The Kapitza resistance of the film-gas interface
(2lb)
If dU /dt is constant for a sufficiently long period all other time derivatives in (14) and (15) may be set to zero. Eliminating Pr between the two equations yields
dU/ dt =(Tr - T0)/ Rr, (22)
where Rr, given by
R r = ( 7rm/2k B) 1;2( To112; Pg)' (23) is the Kapitza resistance of the film-gas interface. In steady state, energy transport between the film and gas is thus linear in the temperature difference.
We should stress that the coefficient of proportionality, R[ 1, depends only on the preexisting gas conditions T0 and Pg. This result, unlike eq. (18) for example, is exact, provided that the gas is largely undisturbed, and valid even if Tr - T0 is not small. The expression for Rr may be compared with previous work by restoring a, the sticking coefficient, to our equations and writing Pg= (Pg/ m)k8T0 • Then,
Rr= (m/2apgk8)(27rm/ k8TQ)112, (24)
which is identical to the result of a simple Knudsen gas model of evaporation from the bulk (24).
To appreciate the role played by Rr in the model let us begin by considering the situations illustrated in figs. I and 3. In most experiments of this kind, Wis not sufficiently small to justify linearizing the dynamical equations. Neverthe- less, some qualitative insight may be gained by doing so. Equating (22) and the linear approximation to (20) combined with ( 18) gives
T1-T0=RrRcW/(Rc+Rk+Rr). (25)
This equation expresses the fact that in steady state as much heat flows into the helium as the gas is capable of carrying away in the form of excess kinetic energy of evaporating over condensing atoms. The situation can be usefully represented by a simple circuit diagram, fig. 6a, in which heat flux appears as electric current, temperature as voltage, and thermal resistance as electrical resistance. The energy flow and temperature drops that result are a conse- quence of the current divider formed by the series combination of R k and R r•
Th -
Re
To
CRYSTAL
D. Goodstein, M. Weimer / Simple model of helium desorption kinetics
w +
dU/dt-
-ThRK Rf
To
FILM-GAS SYSTEM
a
Tf
77R -
dU/dtT1 -,....---~---.-Tt
CRYSTAL
b
Fl LM-GAS SYSTEM
Fig. 6. Equivalent circuits for linearized equations in steady state. Heat fluxes, temperature differences and thermal boundary resistances are equivalent to currents, voltage drops and electrical resistances respectively. (a) Desorption (fig. 1) and power sharing (fig. 3) configurations.
(b) Phonon reflection configuration (fig. 2). Consult text for details.
in parallel with Rc. Now Rr is inversely proportional to the gas pressure and may vary over many orders of magnitude as a function of initial film thickness and temperature. Rk and Rc vary little in comparison. For thin films at low temperatures, when Rr » Rk or Rc, the flow of energy into the helium is effectively choked off. The film temperature approaches that of the heater with very little energy escaping into the gas. A similar circuit (fig. 6b) describes the steady state behavior in reflection experiments, with identical consequences for thin films. If we now consider the opposing limit of thicker films and higher temperatures, R r may become small enough to short circuit R k causing heat to flow freely from the substrate into the gas. The film temperature never gets above ambient and negligible energy is re-radiated back into the crystal. This is basically the explanation offered by Dietsche and Kinder [ 15) for the three-layer effect. Unfortunately, it does not give the three-layer effect quantitatively, and moreover is only applicable in steady state where there is no three-layer effect.
We shall return to these points in section 3.
What we wish to emphasize here instead is that R r is very important in determining the overall character of the film's behavior. This point, illustrated in the steady state above, applies equally well to the full dynamical equations as we shall see below.
D. Goodstein, M. Weimer/ Simple model of helium desorption kinetics
2. 4. The linearized problem Defining
X = ( 8r - 80)/80 ,
(} = (1(- l'o)/ To,
eq. (14) may be rewritten as
R r d(J [ 4 ]
To Q=RrCNdt+(}+RrG (l +8) -1
(26) (27)
+nka8oRr dx
[-/(T o (l
+ 8)) +HI+ 8)- lIn(
Po(To))].
dt k8T0 (l+x)J P8
(28)
We have made use here of the fact that µ1(T) = µsv(T), the chemical potential of the saturated vapor in equilibrium with bulk liquid, where
µsv(T) + Tsi = ~k8T-l(T),
and I is the bulk latent heat of evaporation at temperature T, i(T) = k8T2 d In{ P0(T)] / dT.
(30)
(31) Eqs. (28) and (29) are convenient forms for numerical integration and other manipulations. They also display clearly the importance of Rr, which changes over orders of magnitude while all other coefficients change slowly.
For changes in 8r and Tr small compared to their original values eqs. (28) and (29) may be linearized to leading order in
x
and 8. Then,J = Tb(d8 / dt) - iTaB(dx/dt) + D(}
replaces the expression of energy balance eq. (14), and -Ta( dx/ dt) =Ax+ B(}
replaces the expression of mass balance eq. (15), with coefficients
(32)
(33)
(34a)
D. Goodstein, M. Weimer/ Simple model of helium desorption kinetics
'T8 = 2k8n80Rr, 'Tb= CNRr, (34b)
where P0=P0(T0 ) and 10 is given by eq. (31) evaluated at T= T0. In those instances where the film is adsorbed directly on the heater,
J = Rr Re W (35a)
To Rk+Rc ,
D = 1 + R Rr R , (35b)
c+ k
whereas in the reflection geometry 1 = (Rr/To)11.k,
D =I+ R1/ Rk.
The film is initially undisturbed with x(t = 0) = O(t = 0) = 0.
(36a) (36b)
(37) We are interested in understanding what happens when either a heater is pulsed or a steady phonon flux is suddenly incident at the crystal's surface.
Mathematically, this corresponds to obtaining the solutions of the coupled system, eqs. (32) and (33) subject to (37), for a step in the driving function J. These solutions, conveniently found using Laplace transforms or more elemen- tary techniques, are
0 = Ossf ( t), (38)
x=xssg(t), (39)
with
/(t) = 1-Bi e-1/T, - 02 e-1/T', (40) g(t) = 1 - Xi e-1/T1 - X2 e-1/T2, ( 41) where in steady state
()ss =J/ D, (42)
Xss =-BJ/AD. (43)
One finds that under almost all circumstances the two decay times are widely different in magnitude. Denoting the larger of them 'Ti, then to a very excellent degree of approximation 'Ti » T2 and
( D +
-!B
2 )'T. + A'Tb'Ti= AD (44)
(45)
D. Goodstein, M. Weimer/ Simple model of helium desorption kinetics
The coefficients in ( 40) and ( 41) are given by
81 = (Ar1 -r.)/A( T1 - r2)
=
1 - r./Ar1, 82 = 1 - 81, X1 =r1/ (T1-r2)=1 +r2/T1, x2= l -x1·A circuit whose properties are entirely equivalent to eqs.
illustrated in fig. 7 where Tr - T0 = T08 and with
2nk8S0 2
CL= ~-l=~B .
3~ In( P0/ Pg) '
(46a) (46b) (32) and (33) is
(47) Two time constants appear in the solution, eqs. (38) to (41), because there are two independent heat reservoirs in the problem. Energy may be stored either in elevating the temperature of the film (represented by C N) or in heat of desorption (CL). Energy flowing into the film-gas system is directed through three parallel branches. The current in each path represents, respectively, the heat that goes into warming the film, the addtional kinetic energy of hotter desorbed atoms over colder recondensed ones, and heat of desorption. The net amount of the film desorbed, S0 - Sr= -S0
x,
is given by the voltage drop across CL when all thermal impedances, R,h, of dimension deg cm2 s/W are replaced by mass impedances, Rm, of dimension layers cm2 s/W according to Rm= (BS0/AT0)R,h.In steady state all current into the film-gas system is directed through R r
and the temperature drops across CL and CN are equal. The circuits of fig. 7 then reduce to those shown previously in fig. 6.
w ~
dU/dt
Th -
-
-ThRe RK
To CN R,
CRYSTAL
To
FILM- GAS SYSTEM
a
77R-
T1 -
RK
CL
To CRYSTAL
b
dU/dl
-
FILM-GAS SYSTEM
Fig. 7. Equivalent circuits for linearized dynamical equations (32) and (33) of the text. See eq. (47) for definitions of CL and~. (a) Desorption and power sharing geometries. (b) Phonon reflection geometry.
D. Goodstein, M. Weimer/ Simple model of helium desorption kinetics