A. V. Melkikh and V. D. Seleznev Urals State Technical University, Ekaterinburg
共Submitted December 10, 1998; resubmitted April 22, 1999兲
Pis’ma Zh. Tekh. Fiz. 25, 30–36共December 26, 1999兲
A model is constructed for the transition from evaporative to droplet flow of a liquid through a capillary in a gravitational field allowing for the mutual influence of the droplets. An S-
shaped dependence of the flow on the pressure drop at the capillary is obtained which for certain
共critical兲 values of the control parameter gives a monotonic curve. Values of the pressure
drop are determined for which the droplet flow regime and the droplet-free regime become unstable. It is shown that in a certain range of pressure drops in the presence of noise
transitions may take place from evaporative to droplet flow and back共intermittence兲. © 1999 American Institute of Physics. 关S1063-7850共99兲02112-6兴
The phenomenon of the transition to droplet flow of a liquid in a capillary is used in various devices and techno- logical processes, including inkjet printing, liquid dosing in chemical technology, in medical technology, and so on. The transition to droplet flow has been studied experimentally1–3 and theoretically. Equations describing capillary surfaces for different types of droplet in a gravitational field have been examined in the literature and a linear analysis has also been made of the stability of a suspended droplet.4,5 However, when the droplet has already formed, the droplets may influ- ence each other, which may lead to nonlinear effects. The present paper is devoted to modeling the transition from evaporative to droplet flow allowing for the mutual influence of the droplets.
We shall consider a thin-walled vertical capillary con- taining a liquid. We shall denote by⌬pS0 the pressure drop
of the liquid in the capillary for which a suspended droplet becomes unstable with respect to detachment from the cap- illary. The value of ⌬pS0 can be approximately obtained using the formula
⌬pS0⫽
2
r , 共1兲
where is the surface tension and r is the capillary radius. Quite clearly, after the droplet has fallen, the liquid sur- face at the end of the capillary does not immediately become equilibrium. We shall take to denote the pressure relax- ation time at the free surface and0 to denote the frequency of the ensuing surface oscillations.
Clearly defined oscillations of the liquid surface at the end of the capillary will occur if
Ⰷ0⫺1. 共2兲
For Ⰶ0⫺1 the initial perturbation decays monotoni- cally. The values of and0 may be calculated similarly:6
0⫽
冑
8 r3, ⫽r2
8, 共3兲
where is the dynamic viscosity. An estimate of these val- ues using the formulas 共3兲 shows that for water for example (r
⬘
⬃0.25 mm兲 at room temperature, condition 共2兲 is reliably satisfied.The third characteristic time for this problem is the re- ciprocal dripping frequency J⫺1.
If the perturbation relaxation time becomes compa- rable with the dripping period J⫺1, the stability conditions will differ for the first and subsequent droplets.
We shall make a more detailed study of the case when the liquid surface undergoes oscillations. The amplitude of these oscillations decays exponentially:
⌬x⫽⌬x0exp
冉
⫺ t
冊
,where⌬x0 is the initial perturbation.
At the instant when the droplet has become detached, the liquid surface is elongated by the force mg where m is the droplet mass. The amplitude of the residual pressure p1共as a result of the deformation of the surface after detachment of the droplet兲 will then decay with time as given by
p1⫽gr exp
冉
⫺ t
冊
,where is the liquid density and is a dimensionless coef- ficient of the order of unity.
Quite clearly, the next droplet will occur under condi- tions different from the first. This difference will be that the residual pressure will allow the droplet to fall when ⌬p
⬍⌬pS0.
In this case, the stability condition共1兲 is transformed to give: ⌬pS⫽ 2 r ⫺gr exp
冉
⫺ 1 J冊
. 共4兲TECHNICAL PHYSICS LETTERS VOLUME 25, NUMBER 12 DECEMBER 1999
983
When the pressure difference at the capillary exceeds
⌬pS 共4兲, this causes hydrodynamic transport of the liquid
through the channel. The resulting flow velocity can then be estimated using the Poiseuille formula:
u⫽⫺ r 2
8L共⌬p⫺⌬pS兲, 共5兲
where is the dynamic viscosity of the liquid and L is the length of the capillary.
By averaging formula共5兲 over time over the period of droplet occurrence, we can easily derive an expression for the dripping frequency:
J⫽ r 4
8Lm共⌬p⫺⌬pS兲⫽ 3r
32L共⌬p⫺⌬pS兲, 共6兲 where m⫽4␥r3/3 is the droplet mass, and␥ is a coeffi- cient which allows for the difference between the droplet and capillary diameters.2
Expressions共6兲 and 共4兲 then yield the expression for the dripping frequency: 32␥L 3r J⫽⌬p⫺ 2 r ⫹gr exp
冉
⫺ 1 J冊
.It is convenient to introduce the dimensionless pressure difference and dripping frequency:
⌬p
⬘
⫽ ⌬pgr and J
⬘
⫽J; and omitting the primes we then obtain:32␥L 3r2gJ⫽⌬p⫺ 2 gr2⫹exp
冉
⫺ 1 J冊
, M J⫽⌬⫺B⫺1⫹exp冉
⫺1 J冊
, 共7兲where B⫽ (gr2/2) is the Bond number which character- izes the ratio of the gravitational forces to the surface tension
forces and M⫽ (32␥L/3r2g) is some dimensionless parameter which characterizes the ratio of the viscosity forces to the gravitational forces.
Graphs of J as a function of⌬ are plotted in Fig. 1 for various values of M.
The curve is S-shaped. At points A and B the regime changes. For M⬎Mcthe curve is monotonic and the hyster-
esis disappears. From Eq. 共7兲 by differentiating ⌬p(J): ⌬p J ⫽M⫺ 1 J2exp
冉
⫺ 1 J冊
⫽0 共8兲we can easily find the coordinates of points A and B. These coordinates will correspond to a loss of stability by the sys- tem.
Equation共8兲 can either have two solutions or none at all. The stability boundary of the droplet 关condition 共1兲兴 is as- sumed to be given. Both solutions correspond to the stability boundaries of the droplet regimes, i.e., ‘‘fast’’ and ‘‘slow.’’ In fact, if the droplets influence each other and can fall when
⌬p⬍B⫺1, the flux cannot be infinitely small since the mu- tual influence of the droplets disappears in this case. That is to say, the transition from the droplet regime to the absence of droplets in the region ⌬pS1⬍⌬p⬍B⫺1should take place
abruptly.
Equating the second derivative to zero 2⌬p J2 ⫽exp
冉
⫺ 1 J冊
冋
2 J3⫺ 1 J4册
⫽0and using equality 共7兲, we obtain the parameters of the sys- tem at the critical point:
Jc⫽
1
2, Mc⫽4e
⫺2, ⌬p
c⫽e⫺2⫹B⫺1.
If M⬎4e⫺2 共for example, if the viscosity is high or the channel is long兲, hysteresis cannot occur in the system and the transition from droplet to evaporative flow共in Fig. 1 the line J⫽0 for ⌬p⬍B⫺1) and back will take place continu-
FIG. 1.
ously 共without a jump兲. In this case, the droplets have little influence on each other共see, for example, Ref. 1兲.
If M⬍4e⫺2, hysteresis may be observed in the system: if the pressure drop increases from the point⌬p⬍⌬pS1, the
system abruptly goes over to the upper branch of the curve J(⌬p), whereas if the system moves in the opposite direc- tion from the point⌬p⬎⌬pS2, at the point⌬pS1it will drop
abruptly to zero flux. Thus, we obtain an S-shaped curve for the system being modeled which is typical for example of trigger systems.7
The combination of all the points of stability loss 共for various values of M ) is a curve in coordinates J and ⌬p. This curve is an analog of the spinodal for equilibrium phases.
In order to obtain the spinodal equation in the coordi-
nates J and ⌬p, we express M from Eq. 共8兲 and substitute into Eq.共7兲:
⌬p⫽B⫺1⫺exp
冉
⫺1 J冊冋
1⫺1
J
册
. 共9兲This equation describes a curve, i.e., a set of all the spinodal points 共Fig. 2兲.
In the presence of noise, the behavior of the system changes. Quite clearly, in the region ⌬pS1⬍⌬p⬍⌬pS2 the system is metastable, i.e., both possible states, being stable with respect to small perturbations, are unstable with respect to finite perturbations. In the presence of noise a transition will therefore take place from the lower curve J⫽0 to the upper curve and back. This behavior of the system is called intermittence.
Clearly, the specified interval must include a value ⌬p
⫽⌬pB for which both types of flow are equally probable. A
parallel can be drawn between the two types of flow and the equilibrium phases 共for example, for a vapor–liquid transi- tion兲. The numbers ⌬pS1 and⌬pS2 correspond to the spin-
odals of both phases and ⌬pB corresponds to the binodal.
This work was supported by the RFBR共Grant No. 98- 01-00879兲.
1S. D. R. Wilson, J. Fluid Mech. 190, 561共1988兲.
2A. S. Skotnikov and T. B. Kholina, Khim. Promysh. No. 5, 44共1985兲. 3V. F. Dunaski and N. V. Nikitin, Zh. Prikl. Mekh. Tekh. Fiz. No. 1, 49
共1980兲.
4
A. I. Grigor’ev, A. A. Zemskov, and S. O. Shiryaeva, Nauch. Prib. 1共2兲, 50共1991兲.
5R. Finn, Equilibrium Capillary Surfaces 共Springer-Verlag, New York,
1986; Mir, Moscow, 1989, 312 pp.兲.
6
L. D. Landau and E. M. Lifshitz, Fluid Mechanics,共2nd ed. Pergamon Press, Oxford, 1987; 3rd ed., Nauka, Moscow, 1989, 733 pp.兲.
7W. Ebeling, Strukturbildung bei Irreversiblen Prozessen 共Teubner,
Leipzig, 1976; Mir, Moscow, 1979, 279 pp.兲. Translated by R. M. Durham
FIG. 2.
985
Growth of perfect-crystal Si–Si1
ⴚxGe
x–„Ge2…
1ⴚx„InP…
xstructures from the liquid phase
A. S. Saidov, E´ . A. Koshchanov, A. Sh. Razzakov, and Sh. K. IsmailovPhysicotechnical Institute, Scientific Industrial Association‘‘Solar Physics,’’ Uzbek Academy of Sciences, Tashkent
共Submitted February 12, 1999兲
Pis’ma Zh. Tekh. Fiz. 25, 37–40共December 26, 1999兲
Structures comprising Si–Si1⫺xGex–(Ge2)1⫺x(InP)x with an intermediate Si1⫺xGex buffer layer were grown on silicon substrates. Morphological examinations, scanning patterns and
diffraction spectra, and also the electrophysical and luminescence properties of the heterostructures were used to show that the crystal perfection of these structures depends on the choice of liquid-phase epitaxy conditions. © 1999 American Institute of Physics.关S1063-7850共99兲02212-0兴
It is well known that III–V semiconductors and solid solutions based on them and possessing their unique proper- ties are among the most interesting materials for optoelec- tronics. Consequently, studies of the possibilities of obtain- ing these materials on relatively cheap substrates such as silicon are among the topical issues in semiconductor mate- rials technology.
In Ref. 1 we showed that (Ge2)1⫺x(InP)xsolid solutions
can be obtained on silicon substrates and we described re- sults of some preliminary investigations. In the present paper we report results of investigations to study the possibility of obtaining layers of semiconducting perfect-crystal (Ge2)1⫺x(InP)xsubstitutional solid solutions on silicon sub- strates with an intermediate Si1⫺xGexbuffer layer by epitax- ial growth from an indium flux in a single growth cycle.
The layers are grown from a flux bounded by two hori- zontal substrates using the forced cooling method. Before growing the epitaxial layers of solid solutions, we searched the published data on the solubility of Si, Ge, and InP in various metal solvents and in various temperature ranges in order to select a suitable solvent. Since our flux is a multi- component one, when selecting the composition we had to take into account the influence of intercomponent interaction on their solubility in the liquid phase. Since no such data are available in the literature, we carried out preliminary experi- ments to study the indium corner of the phase diagram of a multicomponent In–Si–Ge–InP system in order to refine the composition of the flux.
Thus, we selected a suitable In–Si–Ge–InP flux compo- sition and temperature range to grow an Si1⫺xGex–
(Ge2)1⫺x(InP)xepitaxial structure on silicon substrates in a
single growth cycle. The structure was grown in the tempera- ture range 700–850 °C and the rate of forced cooling was 1.0–1.5 deg/min.
The substrates were KE´ F-grade (n⫽5⫻1017cm⫺3) and KDB-grade ( p⫽1.1⫻1017cm⫺3) single-crystal silicon wa- fers (d ⫽ 25–30 mm兲 having misorientations of between 0°15
⬘
and 3° relative to the关111兴 crystallographic direction. The thickness of the epitaxial layers varied between 15 and 25m depending on the crystallization initiation tempera- ture and the growth interval, the composition of the liquid phase, the position of the substrates relative to the flux, andalso the rate of forced cooling. Morphological examinations of a cleaved section of the structure and the surface of the epitaxial layers, carried out using an MIM-8M metallo- graphic microscope, showed that other conditions being equal, mirror-smooth epitaxial layers grow on the silicon substrates having the smallest misorientation (0015
⬘
).The structural perfection of the as-grown layers also de- pended on the gap (␦) between the horizontal substrates which could be varied in the range ␦ ⫽ 0.25–2.5 mm by using special graphite supports. The structurally most perfect layers of solid solutions were obtained on the upper and lower substrates for ␦ ⫽ 0.25–0.6 mm. For ␦⬎0.6 mm the quality of the layers grown on the upper substrates deterio- rated significantly. An investigation of the component distri- bution over the thickness of the epitaxial layer determined using a CAMECA microanalyzer showed that when a par- ticular growth regime is maintained, an Si1⫺xGexlayer crys-
tallizes initially on the silicon substrate, beginning with the silicon.
The intermediate Si1⫺xGex layer then gradually gives
way to a variable-gap layer of (Ge2)1⫺x(InP)xsolid solution
whose InP content increases in the direction of growth. The ratios of the Si1⫺xGexand (Ge2)1⫺x(InP)xlayer thicknesses
may be varied depending on the growth conditions.
An x-ray diffraction analysis of the structural perfection of the layers using a DRON-UM1 device showed that the as-grown epitaxial layers of Si1⫺xGex–(Ge2)1⫺x(InP)xsolid
solutions possess extremely good single-crystal properties and extremely low stresses, as is evidenced by the absence of peaks corresponding to phases other than the initial ones on the diffraction pattern. The diffraction spectra were obtained by continuous recording using copper-anode radiation (
⫽1.5418 Å, ⫽1.3922 Å兲. The anode voltage and current
were 30 kV and 10 mA, respectively. The exposure time was varied between 1 and 3 h. We estimated the crystal lattice constants of the substrate and the Si1⫺xGex and (Ge2)1⫺x(InP)x epitaxial layers, which were 5.420, 5.653, and 5.743 Å, respectively共Fig. 1兲.
We investigated the spectral dependence of the photolu- minescence at 77 K obtained from the surface of the epitaxial Si–Si1⫺xGex–(Ge2)1⫺x(InP)xstructures. The edge emission
band corresponded to the band gap of indium phosphide,
TECHNICAL PHYSICS LETTERS VOLUME 25, NUMBER 12 DECEMBER 1999
986
1.34 eV, which indicates that the surface of the epitaxial layer has an InP composition共Fig. 2兲.
We studied some electrophysical properties of films grown on high-resistivity substrates. We determined the re- sistivity, the type of conductivity, and the carrier concentra-
tion at 300 K (⫽0.1–1 ⍀•cm, n⫽2.8–5⫻1017cm⫺3). The films possess n-type conductivity.
To sum up, we have shown that perfect-crystal epitaxial layers of (Ge2)1⫺x(InP)xsolid solutions can be grown from
the liquid phase on silicon substrates by suitably selecting the growth conditions.
1
A. S. Saidov, Dokl. Akad. Nauk Uz. SSR. No. 1, 17共1991兲. Translated by R. M. Durham
FIG. 1. Diffraction pattern of Si–Si1⫺xGex–共Ge2)1⫺x共InP兲x heterostruc-
tures.
FIG. 2. Spectral dependence of the photoluminescence at 77 K obtained from the surface of Si–Si1⫺xGex–共Ge2)1⫺x共InP兲xstructures.
987