- Email: [email protected]
Liquid bridge stability data
Journal of Crystal Growth 78 (1986) 369—378 North-Holland, Amsterdam
369
LIQUID BRIDGE STABILITY DATA I. MARTINEZ and J.M. PERALES Unwersidad Politécnica de Madrid, ETSI 4 eronduticos, 28040-Madrid, Spain Received 8 August 1985~manuscript received in final form 24 July 1986
High precision computations of the minimum volume and the corresponding shape of axisymmetric liquid bridges at rest, anchored to unequal and coaxial discs, are presented. The tables and graphs provide a standard for comparison of linearized and other approximate models, as well an aid in detecting very small departures in shape due to weak (for instance, electrostatic) forces. Relevance to crystal growth by the floating zone technique in the absence of gravity is elucidated.
1. Introduction The configuration shown in fig. 1 is considered, This is a classical setup that has been used since the pioneering work of Plateau in the 19th century. Because in most cases the influence of the protruding solid tips immersed in the liquid is just
a constant shift in the scale of volumes in tables and graphs, the solid/liquid interface is assumed to be planar. The different shapes a liquid bridge can present and the criteria for their stability are well documented [1—5],but the direct problem of finding the shape for a given geometry (R1, R2 and L) and volume (V) is cumbersome in practice, and it is even more cumbersome to find the minimum volume for a given geometry, although helpful approximations are available [6]. It might be argued that time has passed for the publication lowing. Unfortunately, of long datafor tables the such time as being, thosecomfolputing the shape corresponding to a liquid bridge of say 10 and 20 mm in diameter, a distance of 30 mm apart, in the limiting configuration of minimum volume, may take several hours of cumbersome interaction at a computer desk after days of preparatory work, and this burden may be saved to the many crystal-growth investigators interested in floating zone stability. Data are presented here in a very compact 0022-0248/86/$03.50 © Elsevier Science Publishers (North-Holland Physics Publishing Division)
form; all the information to describe a shape is reduced to three numbers. Thus, nearly 200, high precision, singular shapes are documented in this paper. The more complex problem of liquid bridges in a gravity field and/or with rotation has not, so far, been reduced in such a way, although some,
low precision, graphical information is available [7—Il]. The relevance of the data supplied here to crystal growth by the floating zone technique in the absence of gravity may be illustrated by the following example. In recent experiments involving the growth of a silicon rod in space [12,13], one of the two rods processed suffered a bridge disruption. Image recordings seem to indicate that a departure from expected equilibrium shapes already existed, but, how far from the expected
r °
2R2dZ
v=itf R)
~ I
ZR)
0-T~
1
-
2i -
____
-
I.
~—
-
r=cos
a Z
-
Fig. 1. Nomenclature for an anchored liquid bridge at rest, in .
absence of gravity. charactenzed by its geometry Ri. R, and L, and the straight cross-section volume V (liquid plus protruding tips, if any).
B.V.
370
1. Mardncz, .1. 3.1. f’erulcs
liquid hru/gc i ta/nlirr ila~i
stability limit did the rupture occur? The collec-
with ~
<
tion of data supplied here could he of assistance not only for a posterlori analysis hut also during actual growing. Moreover, because small disturbances upon the molten shape are so greatly amplified near the minimum volume stability limit. a fine comparison of actual and predicted shapes could give clues about surface tension gradients (predicted shapes assume constant surface tension), for instance, that may he difficult to detect otherwise (measuring surface temperatures. for instance).
R~
r(a.
q~
2. [he algebraic relationship hetween the three internal parameters a, ~ and q-, and the physical (dimensional) input data R~. R L and V is: R r( (X ~ KaE—~-= ~ (3a) 92) a .%~—~-----— R1 +R-,
=
)
~
:( a
—
(3h)
q~
r(a. ~)+r(a,
-
q.
t. =
(R1
)
i(a.
-
[r(a.
+R~)
~
_____
)]
~)I
~)+r(a.
-
(3c)
2. Formulation For a liquid bridge at rest, in the absence of gravity, surrounded by another fluid, with constant interface tension (constant free energy per unit area), the equilibrium shape is a surface of revolution or, of equivalently, uniform meana surface curvature formulation) of (local revolution
where, besides the functions _~( a. ~) and iC a. q)) already introduced in eq. (1). the function i’( a, ~) that gives the volume of the Plateau curve from q~= 0 up to section ~ ts used [4]: 2 t’(
a.
~)
cosa)
—
r~)( r
~ [r~:(1
=
of minimum effective area (integral formulation). If the liquid is anchored to the disc edges. the effective area is Just the liquid/fluid interface area. The actual meridian curve generating the surface of revolution must be a piece of an axially periodic Plateau curve: cylinder, unduloids, catenary, nodoids and sphere, all of which except the catenary can he expressed in parametric form in terms of the elliptic integrals of first (F) and second (F) class [41:
cos a -I- 2(1 + cos a) L (4) The problem is to pass from the physical variables to the intrinsic shape values a. ~ and q~ by means of eqs. (3). hut solving this system is not an easy task (see appendix 1) and for most applicatio~swith % — ~ I, V— 2~:~ ~ I and Ii ~ I (H = (1 — K )/(I + K ) is used here instead ol’ K. K = (I — H )/(1 + H) for convenience, and H ~ I is equivalent to K 1 ~< I). the direct linear approximation 161.
z(a.~)=cosaF(a.~)+E(a,~),
R~(z.H,,1,v)
r(a,
~)
=
(Ih)
(1— sin2a sin2~)t/2. -
-
(Ia)
-
I + -—
.
were a identifies the Plateau curve (cos a is the ratio of hollow-to-summit radii, fig. 1) and ~ is a
+
~‘
cos(Z — Z~)— cos sin ,%/:~— cos
~ 2~.%
2 H sin( Z
—
Z 1)
parameter that makes the curve being run. The actual shape would be delimited by knowing a and the two extremes ~ and 42~ in nondimensional terms, using (R1 + R2)/2 as unit length.
~
sin :1 ~
—
+H
cos( — ~
Z —
Z~)
.~
‘
may he good enough. Z(a.
~
~t.
~2)_2(
R(a,
~,
~.
~
=
~i)+r(a,
2
r(a,
r(a. ~
~)
+ r(a.
~
.
~2)
(2a)
2.!.
(2b)
Inversion of the system of eqs. (3) gives the parameters to easily compute the shape. with eqs.
StaIn/itt’
1. Martinez, f.M. Perales
(2), for a given geometry. To conduct a stability analysis, the zero-jacobian condition (differential approach, Martinez [1983]) or the conjugate point condition (variational approach, Gillette and Dyson [19711) must be used. The limiting condition for stability, following the former method, is
/
371
Liquid bridge stahiliti’ data
that series. Although they might be approached asymptotically, their importance deserves a special analysis. These shapes are catenoidal, that is, portions of r = cosh z with the appropriate scaling. One of these equilibrium catenoids lies on the minimum volume limit and thus must be found
4~ ~2) ~K/aa aK/a~,
=
~A/8a 8A/~~
OK/8~,
8A/a~2
OV/0a 8V/0~
Table 1 Nondimensional minimum volume V. for liquid bridges at rest,
1I
=
0.
(6)
BV/0~5
For K and A fixed, the system formed by eqs. (3) and (6) consists of four equations with four unknowns, is a, chosen 4~,42 (the and equations V. which, ifare themultivalued), appropriate solution give the minimum stable volume, [ç~. A linear approximation [4] may be used advantageously for long bridges between slightly different discs: 19 ~ain
=
2A (2A
—
—
~
i
—
H J2/3 sin A/A ‘ (7)
2.2. Special cases All curves and graphs presented in this paper correspond to points that are singular in the transformation K, A, V~-sa, 4’~ ~2 according to eq. (6), but we refer now to singularities that remain in the computation after eq. (6) is accounted for, They correspond to the catenoidal stability limit (already mentioned in section 1 when mentioning the catenary as a special meridian curve), to the stability limit with zero slope (dr/dz = 0) at the larger disc, and to the stability limit of minimum Plateau undulation (minium a for all Vmjn at constant K; see table 1). Table 1 gives Vmjn versus A for various disc ratios K, and the internal parameters to compute the actual shape by eq. (2), as explained in appendix 3 2.2.1. Catenoidal limit For every disc ratio K, there is one series of shapes not dealt with in eq. (1) because it becomes singular for a = ir/2 that would correspond to
anchored to the edges of coaaial discs of radius
R1
and
R2.
as
a function of nondimensional disc separation \, and diameter ratio K; the parameters a and ~ serve to compute the interface shape (appendix 3): the liquid/solid angle at both ends. ~ and 9, is also presented to check for limiting contact angle conditions 0i
~
1.2504
1.3502 1.0526 0 8061 0.6018 0.4316 0.2882
24.7 41.0 540 64.3 72.4 78.8
24.7 41.0 540 64.3 72.4 78.8
5.984 7.118 8.372 9.915 11.855 14.270 17.245 20.871 25.247 30.477 36669 43.937
1.2075 1.1664 — 1.1002 —0.9770 —0.8346 —0.6620 —0.4253 0.2692 —t).5427 —0.6948 —0.8025 —0.8851
t).1660 0.0605 t).0000 0.0000 0.0000 0.0000 0.0000 1.5708 1.5708 1.5708 1.5708 1.5708
83.9 87.9 9t).0 90.0 90.0 90.0 90.0 90.0 90.0 90.0 90.0 90.0
83.9 87.9 90.0 90.0 9t).0 90.0 90.0 90.0 90.0 90.0 900 9t).t)
06 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2
1115 1.762 2.462 3.240 4.119 5.131 6.328 7.778 9.557
—1.5235 —1.4567 — 1.3970 — 1.3420 — 1.2885 — 1.2322 — 1.1689 — 1.0985 — 1.0263
1.3667 1.0903 0.8691 0.6993 0.5799 0.5124 0.4950 t).5188 0.5763
246 39.9 51.8 60.7 67.1 71.2 73.6 74.8 75.6
246 41.8 55.6 66.8 75.8 83.0 88.6 92.7 95.9
2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8
11.749 14.440 17.724 21.698 26.466 32.136 38.823 46.646
0.9608 —0.9129 —0.8920 —0.8990 —0.9260 0.9630 — 1.0029 — 1.0418
0.6649 0.7807 0.9113 1.0366 1.1420 1.2239 1.2857 1.3323
76.1 76.5 76.8 77.1 77.3 77.5 77.7 77.9
98.3 100.3 101.9 103.3 104.4 105.4 106.2 107.0
.1 K =
0.6 0.8 1.0 1.2 1.4 1.6
1
1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 K =
V
a
1.107 1.744 2431 3.187 4.025
—
1.5231 1.4567 13978 —1.3447 —1.2959
4,955
—
0.9
1. Martinez, .1. M. Perales / liqiod bridge stahl/ui’ ul~tui
372
Table 1 (continued)
Table 1 (continued)
S
.5
I’
K = ((.8 )).6 1.142 0.8 1.821 1.0 2.564 1.2 3.408 1.4 4.391 1.6 5.566 1.8 6.994 2.0 8.748 2.2 10.910 2.4 13.566 2.6 16.810 2.8 20.742 3.0 25.468 3.2 31.097 3.4 37.748 16 45 540 K = 0.7 (.6 1.194 ((.8 1.931 11) 2.752 1.2 3.701 1.4 4.834 1.6 6.212 1.8 7.906 2.0 9.996 2.2 12.57)) 2.4 15.723 2.6 19.556 2.8 31) 3 2 3.4
24.178 29.70)) 36.243 43.93))
A’ ().6 (1.6 1.278 (1.8 11) 1.2 1.4 1.6 1.8 2.0
2.108 3.043 4.142 5.470 7.11)1 9.114 11.600
o 1.5251 1.4568 — 1.3951 1.3365 —-1.2780 1.2178 — 1.1569 -—1.0988 1.0500 — 11)165 — 118)19 1.0054 -- 1.0224 --1.0474 - 11)760 —11052 — -
1.3921 1.1417 0.9510 0.8177 0.7383 0.7066 0.7147 0.7553 0.8220 0.9071 1.0001 11)901 1.1693 1.2348 1.2876 1 3295
24.4 38.5 49.3 57.1 62.4 66.)) 68.3 69.9 71.1 72.0 72.7 73.3 73.8 74.3 74.7 75 1
24.2 42.2 56.7 68.2 77.4 84.7 90.4 951) 98.6 101.6 104.1 106.1 107.9 11)9.5 110.8 112))
1.5277 — 1.4573 — 1.3932 1.3323 -- 1.2731 --1.2161 1.1645 --1.1223 -— 1.0936 — 1.0803 ~- 1.0811
1.4235 1.2019 1.1)411 0.9373 0.8836 1)8713 1)8924 0.9387 1.0018 11)725 1.1425
24.6 37.5 47.1 54.1 59.0 62.5 651) 66.8 68.3 69.4 7)).4
23.5 42.2 57.1 691) 78.5 861) 92.1 971) 101.)) 104.4 107.3
—
1.0928 1.1113 1.1335 1.1570
1.2059 1.26)32 1.3053 1.3422
71.2 71.9 72.5 731)
109.7 111 8 113.6 115.2
—
1.5307
1.4551
25.7
22.7
1.4581 --1.3926 -- 1.3317 --1.2757 — 1.2265 1.1872 ‘- 1.1605
1.2642 1.1314 11)517 11069 1.0177 1.0452 1.0901
37.1 45.8 52.3 56.9 60.4 62.9 651)
421) 57.4 69.6 79.4 87.3 93.8 99.1
—
—-
—
2.2 14.655 ‘ 1.1472 1.1436 66.6 103.5 2.4 18.384 -- 1.1461 1.1984 67.9 0(7.2 2.6 22.895 1.1543 1.2493 691) 11)1.4 2.8 28.307 1.1686 1.2941 69.9 113.1 31) 34.740 - 1.1864 1.3321 70.7 115.4 3.2 42.323 — 1.2056 1.3638 71.4 117.5 _____________________________________________________
I’
K = 0.5 0.6 1.408 0.8 2.374 1.)) 3.471 1.2 4.772 1.4 6.357 1.6 8.311 1.8 10.725 2.0 13.700 2.2 17.345 2.4 21.772 2.6 27.104 2.8 33.467 3.0 40.992
o
~i
.
1.5327 1.4591 1.3940 1.3364 1.2874 1.2493 1.2232 1.2098 1.21)73 1.2131 1.2244 1.2388 1.2547
1,4808 1.3233 1.2180 1.1601 1.1409 1.15)13 1.179)) 1.2177 1.2593 1.2993 1.3351 1.366)) 1.3921
28.) 38.)) 45.7 51.6 56.)) 59.3 62.)) 641) 65.7 67.1 68.2 69.2 7)1,1
22.1 42.0 57.8 70.5 80.7 89.)) 95.7 101.4 106.1 10).! 117.5 116.4 119.))
A = (1.4 1)) I 616 )).8 2.775
--
1 S114 1.4599
1 497)) 1.3765
3”— 4)).5
42.7
11) 1.2 1.4 1.6 1.8 21) 2.2 2.4 2.6 2.8
4.098 5.677 7.607 9.987 12.923 16.530 20.928 26.245 32.612 40.168
1.3989 — 1.3488 1.3107 — 1.2856 1.2719 — 1.2686 —1.2727 --1.2816 1.2933 1.3)163
1.3000 1.2631 1.2566 1.2702 1.2958 1.3254 1.355)) 1.3821 1.4059 1.4262
47.1 52.3 56.4 59.5 621) 641) 656 671) 68.2 69.2
58.9 72.)) 82.5 91.1 98.2 11(41) 11(91) 113.2 116.8 1193)
(1.6 ((.8 11) 1.2 1.4 1.6 1.8
1.976 3.405 5.048 7.019 9.430 12.398 16.047
1.5252 1.4617 1.4108 -~1.3732 1.3489 1.3362 — 1.3327
1.5057 1.4254 1.3786 1.3607 1.3628 1.3765 1.3953
38.7 451) 50.3 54.7 58.1 60.9 63.1
24.7 44.9 61.3 74.5 85.2 941) 101.3
21) 2.2 2.4 2.6
20.509 25.919 32.418 40.154
1.3357 1.3427 ‘— 1.352)) 1.3623
1.4151 1.4337 1.4502 1.4645
64.9 66.5 67.7 68.8
107.3 112.4 116.8 12)).5
1.517))
1.5155
47.3
29.9
1.471)3 1.4367 1.4157 1.4063 1.4025 1.4047 1.408) 1,4169 1.4244
1.4748 1.4548 1.4505 1.4544 1.4653 1.4762 1.4869 1.4965 1.5049
51.7 55.6 58.9 61.1 63.7 65.6 67.1 68.4 69.4
49.5 65.6 78.8 88.3 98.2 11(5.5 111.6 116.7 121.1
-
— . —
A = (1.3 —
—
.
A = ((.6
2.656
((.8 11) 1.2 1.4 1.6 1.8 21) 2.2 2.4
4.475 6.592 9.142 11.852 16.092 20.777 26.467 33.318 41.488
--
-‘ --
— —
I. Martinez. f.M. Perales
V
9-
a
4 105
—
1 5214
1 5386
58 7
39 9
0.8
6.536
—
1.5002
1.5279
61.1
1.0
9.416
-
1.2
12.910
1.4876 1.4819
1.5244 1.5252
63.4 65.4
58.3 73.5 86.0
1.4 1.6
17.182 22.400
1.8 2.0
28.737 36.365
1.4808 1.4825 1.4857
1.5284 1.5324 1.5363 1.5400 1.5432
67.2 68.6 69.9 70.9 71.8
96.2 104.6 111.6 117.4 122.3
2.2
45.462
373
Liquid bridge stability data
the general case in eq. (6), and is analyzed apart
Table 1 (continued) A
/
—
—
1.4897 —
1.4939
because the following peculiar property [11]:K) it splits theofminimum volume curve (for constant in two regions, such as for longer bridges the neck travels towards the larger disc during the breaking process, whereas it narrows without axial shifting
for shorter bridges. In this case ~,,
(8a)
=
— =
—arcsin(s/i
2 /sin a),
(8b)
K
—
=J(a) =0. separately (for every disc ratio K) as presented in table 2. This limiting case has already had direct application in the study of disjoining pressure effects on thin liquid films on a solid surface [14], due to its relevance as the most sensitive liquid bridge configurations (the pressure jump across the interface is zero). Exact knowledge of these shapes is required to evaluate their effect when very small forces show up [14], so that, although the computations in this case are not so involved as in the other cases, the explicit form of the determinant corresponding to eq. (6) is presented in appendix 2.
2,2.2. Stability limit with cylindrical ending (dr/dz =
K 1.0 0.9 0.8
results are presented in table 4.
when solving eq. (6), and the
Table 3 Minimum volume shape with cylindrical ending (dr/dz
0~ 12.2 11.7 11.4 11.3 11.2 11.2 10.8 9.8
K 1.0 0.9
a
1.1407 —1.1476
~i
0.0000 0.4984
A 2.1283 1.8620
0.8 0.7
—1.1618
0.7128
—
1.1825 1.2099
0.8813 1.0257
0.5071 0.5017 0.4747
12.2 13.0 14.3 16.2 19.0 22.8 27.9 34.6
—
1.2447
0.4118 0.2869
1.0878 0.7996
43.5 55.5
7.9 4.7
0.2 0.1
1.2880 — 1.3410 — 1.4049 — 1.4810
0.2 0.1
—0.9201 —0.6433
2.6721 3.1881
0.4718 0.4738 0.4801 0.4900 0.5007
=
0) at
the larger disc; for every disc ratio K, the separation A and the corresponding volume V~,, are given: with K, a and ~, the actual shape may be computed as explained in appendix 3. these points correspond to curve B in fig. 2 0i ~:
Oi
0.3
0.4
a
V 0.7129 0.7218 0.7517 0.8058 0.8842 0.9801 1.0761 1.1354
A
0.5
has been asymptotically approached numerically by reducing
2.2392 2.2823 2.3079 2.3172 2.3180 2.3250 2.3599 2.4552
0.6
2,2.3. Stability limit with minimum undulation As mentioned above, the equilibrium shape of an axisymmetric liquid bridge in zero gravity corresponds to a piece of an axially periodic curve (Plateau curve) that can be identified by the hollow/summit radii ratio (cos a), as shown in fig. 1. For every disc ratio K, there is a minimum volume stability limit with a minimum value of a; it
This is really a simplification with respect to
—2.2392 —2.1745 —2.0791 — 1.9501 — 1.7893 — 1.6015 — 1.3926 — 1.1658
0.7
Results are shown in table 3. For the special case of equal discs (K = 1), eq. (8c) reduces to 2E(a, IT/2) — F(a, ~r/2) = 0, which gives A = 2.128392, an inportant value that has appeared with different degrees of error in the literature.
0) at the larger disc
Table 2 Catenoidal stability limit; for every disc ratio K. a separation exists, such as the minimum stable volume V,,,~.,is a catenoidal shape stretching from z 1 to z, when scaled with the neck radius (that is, r = cosh z): 9~and 0, are the angles the liquid forms with the discs: these points correspond to curve C in fig. 2
(8c)
0.6 0.5 0.4 0.3
—
—
—
V
7.901 6.748
90.0 74.0
90.)) 90.0
1.7832
6.862
68.1
90.0
1.7270 1.6778
7.246 7.834
64.2 61.4
90.0 90.0
1.1535
1.6284
8.623
59.8
90.0
1.2677 1.3689 1.4561 1.5264
1.5726 1.5036 1.4111 1.2739
9.630 10.890 12.454 14.389
59.1 59.6 61.7 66.2
90.0 90.0 90.0 90.0
1.
374
1.’lartine:. .1. ¶.,J• Pi,u’o/u’~
liquid (‘ridge ta/ui/u ii’ ,/ata (a)
Table 4 Minimum solume corresponding to minimum undulation ( minimum): for ever\ A’ there is a minimum us ersus ‘I (see table 1) and it is presented here: these points correspond iii curse A in fig. 2
0.1
20
0.5
-,‘
-.
‘ ~‘ ~,‘
V ,
/
o ((.0000 (1.8913
(11(000 ((94(17
31416 S 2845!
I 739 19 18.556
90,0 ~i 76.9
6~ 901) 102.2
((.8 ((.7
1.01(12 11(790
1.0261 1 1028
2.656! 2.4851
17.839 17.265
72,9 69.9
104.7 !((5 7
((.6 (1.5 ((.4 (1.3 (12 ((.1
1.1452 I 207)) 1.2685 1.3327 1.4()25 1.4808
1.1763 1.2492 1.3221 13948 1.4650 1 5280
2.3185 2 1513 1.9778 I 7949 1.5945 1.381)1
16,777 16.389 16.1)93 15.944 15,975 16.717
67.4 65 3 63 8 63! 63 7 67.0
105.8 1(15,)) 103.4 101.1 9S1)
‘/
// ‘
/
10
~/
-
/
/‘
=
‘~
1
—~
‘ ,
/
-
,f
‘‘
_____________________________________________________________________
A’ 1)) ((.9
K
‘
/
1
/
/
~
/
/
‘
/
~
N/s
/
/
‘
/ //~/
5
/
/
,~
/
.
,
~)
/
/
/5
/~‘
0I
NT
0
3. Results and conclusions
1
2
A
3
(b) 20
The minimum volume for stability, versus bridge length, is presented in fig. 2 for several disc ratios K. The special cases described above have
S
‘
\
\~
‘
\
S
,
\
~
31
~-‘-~ ‘
S
.
28
V
been explicitely included. High precision values can be found in the tables, The method to conipute the limiting shapes for the points included, as well as how to extend the tables here given, are As an example, let us consider the bridges that presented in appendix 3. can be formed between discs of R 1 10 mm and R, = 20 mm in diameter (R1/R, = 0.5). assuming a certain liquid having some 30° contact angle with the solid discs in the presence of the outer fluid. A short liquid bridge of L — 9 mm in length (A =0.6) can 3.6) easily established by putting V= If 3 (U— of he liquid in between (fig. 3a). 1.5 cm the liquid is now gently withdraw at constant disc
\\\\
~\‘s\\’
S
.
‘
\ss”
“
\
‘~ \,
1
“ N N
~‘
10
~“=
-
N
~‘
~
\*
N,,
N
~
~
‘—id~-..
-.-
—.
=
~
C
0.1 =<_—~-
0
-
-s~ ‘
=
~-
— --—~—~
0.5
.
______ K
separation. the liquid border will detach from the
1’i~.2. Minimum volume F
disc edge when the angle at the edge of the larger disc diminishes to 30° (fig. 3h). well before the minimum volume V — 1 .4 given in table 1 is reached (fig. 3c). Assume now that one manages to have a long 3 liquid bridge 30 mm = 2) containing 9 cm of liquid (V=of21.3. fig.(A3d); when will such a
euiavial discs of radii R1 and R2 a distance / apart. 1 sr every A’ ‘— R1 R there are three special points in the I, ( .5 curve: )A( bridge of minimum undulation: (B) bridge of dr d:
for a liquid bridge 5inuliored
bridge break if stretched at constant volume’! (fig. 3f). or. alternatively, when will it break if liquid is removed at constant disc separation? (fig. 3e). Answers to these and similar questions are now
o
(cylindrical ending) at the larger disc: ((‘1 caienuuidal
bridge (a)
1) for constant A: )h) ~
(K (for constant
easily obtained from table 1 (or fig. 2). For K = 0.5 and V= 21.3 we see in table 1 that ~ —2.4 (36 mm long), whereas for K = 0.5 and .1 — 2 the minimum volume is V — 13.7 (5.8 cm3). In both cases. 0, being greater than 90°, the breaking
I. %!arone:, f.M. Peru/es
/
375
I.iquud bridge stability data
In conclusion, this paper presents (as far as we
know for the first time) a comprehensive and highly compact data set on the minimum stable
volume for liquid bridges between non-equal discs in absence of gravity; and not only the value of ---I
actual Vmin ~5shape gi~’~’~, of I~~1t the data freenecessary surface in to this compute limiting the configuration is provided.
~ -
Appendix I. Shape computation for given .
K,
A
and
V
Several methods have been investigated to compute the shape from primary data K, A
~
-
-
N
I Fig Fxample of liquid bridge shapes for Ri — 10 mm, R, —‘ 2)) mm (a) Stable bridge with .S — 1)6 and V= 3.6. computed with eq. (9). (h) While removing liquid from the (a) shape, the liquid would recede form the disc edge if the liquid/solid limiting contact angle is assumed to he 30° (see section 3). (c) Minimum volume stability limit for K = 0.5 and A = 0.6. computed as explained in appendix 3. (d) Stable bridge with A = 2 and V= 21 3, computed with eq. (9). (e) Minimum volume stability limit for K 0.5 and A = 2. computed as explained in appendix 3. (f) Stability limit when stretching from (d) at constant volume,
and V. In
all cases, the problem being highly nonlinear, a fairly good initial guess is required; eq. (5) is an If an algebraic solution of eq. (3) is desired, we
evaluable help in that respect. recommend the specification of a and computation ~ and ~2 from the first two conditions in eq. (3); the first one may be solved explicitly as /, = ~,(K, a, ~) and the second one A(a, ~ 4~)= A by a standard root finder, thus obtaining a V = V(a) relationship. Another approach is by a series expansion in terms of orthogonal polynomials, for instance Tschebyscheff economization series. Thence, using L/2 as unit length, the shape r = r( z) can be
written as N
shape (readily obtained from have a hollow and a summit.
K,
a and 4~)will
As regarding the relevance of this results to the real floating zone problem, it might be argued that
r(z)
n=0
with theNnth order Tschebyscheff polynomial, where7 the ± 1 coefficients a, 1 are to be found from the conditions r( —1) = R1/L, r( + 1) =
true molten zones do not have fixed volume or length as the model assumes, but the mchanical stability of a real floating zone (be it a melt or a solution) to have much smaller characteristic happens time scale that athermal or physico-
R2/L.
chemical instabilities and thus, for small time intervals, it is at all licit to treat the liquid volume and length as constants, as has already been suc-
and
cessfully made in [15], where this same liquidbridge model has been used to predict, with high
accuracy, the outer-shape evolution (stable and unstable) of a real floating zone of molten silicon aboard an orbiting laboratory.
a,
=
2 (1 r
f
dz
v =
—
1
r~/1+ (dr/dz)2 dz
=
minimum
for every possible variation of the coefficients. If the series is truncated at N = 3, a 0, a1 and a2 can be explicitly solved for and the minimum of the
376
I. .i’larri,iu’:, .1. .%/. Peso/eu
l.iquid bra/ge siahuliti’ datut
last integral with a~ easily found. Unfortunately this expedient procedure gives an accuracy of the same order as the straightforward expression of eq. (5). As another approach, a very accurate solution can be found by direct integration of the differential equations for constant mean curvature (Laplace capillary equation) in the form
•t’(
dr
i’(:~)=0,
(l3h)
fv(:) cosh : d: =0.
(l3c)
.
with
rI, 1 =1—H --
dt
I +
~2
‘
—~(l +
,.
=
t-)
‘
with
(Jacobi condition in the variational formulation) is 2i’ d t’ d ~ —2 tanh :-~- +p cosft: =0. (12) where p is an internal constant that, in addition to the two boundary conditions in eq. (12), is to he found from the three conditions
‘I~
(9a)
/
=
t~.
)
.~
-
-
which, upon substitution of the general solution of eq. (12) takes the form
9h )
dt’
with
~r
=
“I
=
sinh
where the internal variable t (the local slope of the shape) and the nondimensional pressure jump aceross the free surface p (constant) are introduced. In this scheme H, H = (1 — K)/(1 + K), is given, and p and t~ are estimated with the aid of eq. (5). A standard Runge—Kutta numerical integration of eq. (9) from : = 0 to : = 2A (.‘~ is given) then yields new values for H and V =
r
I
V = (1
~
—1.
(lOa)
8’ I =~ —
H
+
r
)3
.1/
0. (9c)
H
( l3a)
0.
=
(lOb)
and a standard Newton—Raphson iteration is used to approach the specified H and V to the desired accuracy. The procedure, of course, becomes singular near the stability limit. Appendix 2. Stabilit~ limit corresponding to a catenoid Quasi-catenoidal shapes can be expressed. when properly scaled, as
(cosh 2:,
I)
sinli cosli
uosh’1
siili coshi
costs’:,
/) :~(
ZS(
g(
:~)
g)
I
cosh 2:~(/4 ‘
where the functions -
f(:)
=
—
g(:)
=
~-
f(:) and
.
sinh(2:)
—
+
+ ~ sinh(2:) + .
g(:) are
(14)
-
~-
.~,
cosh(2: ). sinh(4:).
(ISa) (15h)
‘
After substitution in eqs. (14) and (15) ol cosh :, = arg cosh K ,
(16)
the only remaining unknown is Zi. which is to he found from a nontrivial solution of eq. (14). After is found the nondimensional bridge length and liquid volume are given by 2 — t , (l7a) cosh :, + cosh sinh(2z, ) + 2:~ — sinh(2: ) — 2: V=2 (17h) (cosh :2 + cosh :~) These values can he found in table 2 as a funtion =
of disc ratio K. Appendix 3. Summary of computational procedures
r(:)=cosh:+s’(:).
(11)
where is a small number. After substitution in the equilibrium equation (Laplace capillary equalion, or Euler—Lagrange equation in the variational formulation) the resulting stability equation
How
to find
the shape for points in tab/es’ 1. .1 and 4
(1) Given K. a and ~. (2) Conipute ~ = ~( K, a, /~) from eq. (3a): function ~, below.
I. Martinez, f.M. Peru/es
(3) Compute the shape z = z(a, ~, ~ ~, r= r(a, ~, 4~,~); PROCEDURE Shape below.
/
377
Liquid bridge stability data
How to find new points
(3) Interpolate a(K, A) from table 1. (4) Compute 4~= ~ 1(K, A, a) from eq. (3b); function 4~below. (5) Compute Vmjn with eqs. (3c) and (4) and the
(1) Given K and A. (2) Compute AB(K) from eq. (7) or points in curve B, fig. 2.
shape (z, r) as above. (6) To check or enhance accuracy, evaluate the jacobian in eq. (6), slightly change a, and repeat steps 4 to 6.
Pseudocode for the computer routines mentioned above
FUNCTION r
=
r(cL,~)
(1_sin2asin2~)~2
FUNCTION ~2(K, a, 2 =
~
±arc sin VI1~r(d1,~1)/K) sin a
use + if
A
-
if A>AB
PROCEDURE Shape(a,~ 1,42,z,r,n)
! z & r are vectors of size n
for i0 to n ~~+( qa,- q~)(i/n) P1ateau~cs,i~,z(i),r(i)) next i
FUNCTION ~1(K,A,cs) =
always 0<~
arc sin S ~flO
loop ~2=qi2(K,
a,
~)
Plateau(cs, P1,z1,r1) Plateau(a, 1~9,z2,r2) NewA’~(z2-z1)/(r2+r1) repeat ioop, changing ~ until NewAA
PROCEDURE Plateau(cs, ~,z,r) Elliptic integrals(ct,~,F,E) zcosruF+E rr( a, qi)
! standard library routine
1. .3-lartine:. .1. 1.1. Pera/e,v
378
,/
Luquid bridge sw/u/dr i/ala
IXI
References
L.A. B&uueher and M.J.B. Evana J. (‘olloid Interface Sc). 75 (1980) 409.
I~IR.A.
III
RD. (rillette and l’).C 196
121
SR. (one]) and (1977) 466.
131
A. Sanz and I. Martinez, J. (‘olloid (1983) 235
141
I. Martinez. in: Material Sciences under Microgravity. ESA SP-191 (ESA, Paris. 1983) PP 267—273
1131
151
J. Mcsegucr, J. (‘rystal Growth 67 (1984) 141. J. Mcseguer. in: Material Sciences cinder Microgravitv, ESA SP-222 (ESA, Paris. 1984) pp 297- 30)(
under Microgravitv. ESA SP-222 (ESA. Paris. 1984) pp. 173-- 181. 1141 J.F. Paddav, in: Material Sciences cinder Mucrogras’it\.
I (21
I~I SR.
Dyson, (‘hem. Eng. J .3 (1972)
MR. (‘ordcs, J. Crystal Growth 42 Interface Sc). 93
Coriell. S.C. Hardy and MR. Cordes..l. Interface Sc) 60 (1977)126
Brown and L.L. Scruvcn, Phil. Trans. Roy. Soc. london A297 (1980) 51
1101 1.1-1. l.Jngar and R.A. Brown, Phil. London A306 (1982) 347.
Trans. Roy. Soc.
Ill J. Meseguer and A. Sanz. J. Fluid Mccli. 153 (1985) 83. {12I A. Eyer. B.O. Kolhesen and R. Nitsche. 3. Crystal Growth
(‘olloid
1151
~7 (1982)145. A. Ever. H. Leiste and R. Nitsehc, in: Material Sciences
ESA SP222 (ESA, Paris, 1984) pp. 9-14. I. Martinez and A. Eyer, J. Crystal (irowth 750986) 535.
369
LIQUID BRIDGE STABILITY DATA I. MARTINEZ and J.M. PERALES Unwersidad Politécnica de Madrid, ETSI 4 eronduticos, 28040-Madrid, Spain Received 8 August 1985~manuscript received in final form 24 July 1986
High precision computations of the minimum volume and the corresponding shape of axisymmetric liquid bridges at rest, anchored to unequal and coaxial discs, are presented. The tables and graphs provide a standard for comparison of linearized and other approximate models, as well an aid in detecting very small departures in shape due to weak (for instance, electrostatic) forces. Relevance to crystal growth by the floating zone technique in the absence of gravity is elucidated.
1. Introduction The configuration shown in fig. 1 is considered, This is a classical setup that has been used since the pioneering work of Plateau in the 19th century. Because in most cases the influence of the protruding solid tips immersed in the liquid is just
a constant shift in the scale of volumes in tables and graphs, the solid/liquid interface is assumed to be planar. The different shapes a liquid bridge can present and the criteria for their stability are well documented [1—5],but the direct problem of finding the shape for a given geometry (R1, R2 and L) and volume (V) is cumbersome in practice, and it is even more cumbersome to find the minimum volume for a given geometry, although helpful approximations are available [6]. It might be argued that time has passed for the publication lowing. Unfortunately, of long datafor tables the such time as being, thosecomfolputing the shape corresponding to a liquid bridge of say 10 and 20 mm in diameter, a distance of 30 mm apart, in the limiting configuration of minimum volume, may take several hours of cumbersome interaction at a computer desk after days of preparatory work, and this burden may be saved to the many crystal-growth investigators interested in floating zone stability. Data are presented here in a very compact 0022-0248/86/$03.50 © Elsevier Science Publishers (North-Holland Physics Publishing Division)
form; all the information to describe a shape is reduced to three numbers. Thus, nearly 200, high precision, singular shapes are documented in this paper. The more complex problem of liquid bridges in a gravity field and/or with rotation has not, so far, been reduced in such a way, although some,
low precision, graphical information is available [7—Il]. The relevance of the data supplied here to crystal growth by the floating zone technique in the absence of gravity may be illustrated by the following example. In recent experiments involving the growth of a silicon rod in space [12,13], one of the two rods processed suffered a bridge disruption. Image recordings seem to indicate that a departure from expected equilibrium shapes already existed, but, how far from the expected
r °
2R2dZ
v=itf R)
~ I
ZR)
0-T~
1
-
2i -
____
-
I.
~—
-
r=cos
a Z
-
Fig. 1. Nomenclature for an anchored liquid bridge at rest, in .
absence of gravity. charactenzed by its geometry Ri. R, and L, and the straight cross-section volume V (liquid plus protruding tips, if any).
B.V.
370
1. Mardncz, .1. 3.1. f’erulcs
liquid hru/gc i ta/nlirr ila~i
stability limit did the rupture occur? The collec-
with ~
<
tion of data supplied here could he of assistance not only for a posterlori analysis hut also during actual growing. Moreover, because small disturbances upon the molten shape are so greatly amplified near the minimum volume stability limit. a fine comparison of actual and predicted shapes could give clues about surface tension gradients (predicted shapes assume constant surface tension), for instance, that may he difficult to detect otherwise (measuring surface temperatures. for instance).
R~
r(a.
q~
2. [he algebraic relationship hetween the three internal parameters a, ~ and q-, and the physical (dimensional) input data R~. R L and V is: R r( (X ~ KaE—~-= ~ (3a) 92) a .%~—~-----— R1 +R-,
=
)
~
:( a
—
(3h)
q~
r(a. ~)+r(a,
-
q.
t. =
(R1
)
i(a.
-
[r(a.
+R~)
~
_____
)]
~)I
~)+r(a.
-
(3c)
2. Formulation For a liquid bridge at rest, in the absence of gravity, surrounded by another fluid, with constant interface tension (constant free energy per unit area), the equilibrium shape is a surface of revolution or, of equivalently, uniform meana surface curvature formulation) of (local revolution
where, besides the functions _~( a. ~) and iC a. q)) already introduced in eq. (1). the function i’( a, ~) that gives the volume of the Plateau curve from q~= 0 up to section ~ ts used [4]: 2 t’(
a.
~)
cosa)
—
r~)( r
~ [r~:(1
=
of minimum effective area (integral formulation). If the liquid is anchored to the disc edges. the effective area is Just the liquid/fluid interface area. The actual meridian curve generating the surface of revolution must be a piece of an axially periodic Plateau curve: cylinder, unduloids, catenary, nodoids and sphere, all of which except the catenary can he expressed in parametric form in terms of the elliptic integrals of first (F) and second (F) class [41:
cos a -I- 2(1 + cos a) L (4) The problem is to pass from the physical variables to the intrinsic shape values a. ~ and q~ by means of eqs. (3). hut solving this system is not an easy task (see appendix 1) and for most applicatio~swith % — ~ I, V— 2~:~ ~ I and Ii ~ I (H = (1 — K )/(I + K ) is used here instead ol’ K. K = (I — H )/(1 + H) for convenience, and H ~ I is equivalent to K 1 ~< I). the direct linear approximation 161.
z(a.~)=cosaF(a.~)+E(a,~),
R~(z.H,,1,v)
r(a,
~)
=
(Ih)
(1— sin2a sin2~)t/2. -
-
(Ia)
-
I + -—
.
were a identifies the Plateau curve (cos a is the ratio of hollow-to-summit radii, fig. 1) and ~ is a
+
~‘
cos(Z — Z~)— cos sin ,%/:~— cos
~ 2~.%
2 H sin( Z
—
Z 1)
parameter that makes the curve being run. The actual shape would be delimited by knowing a and the two extremes ~ and 42~ in nondimensional terms, using (R1 + R2)/2 as unit length.
~
sin :1 ~
—
+H
cos( — ~
Z —
Z~)
.~
‘
may he good enough. Z(a.
~
~t.
~2)_2(
R(a,
~,
~.
~
=
~i)+r(a,
2
r(a,
r(a. ~
~)
+ r(a.
~
.
~2)
(2a)
2.!.
(2b)
Inversion of the system of eqs. (3) gives the parameters to easily compute the shape. with eqs.
StaIn/itt’
1. Martinez, f.M. Perales
(2), for a given geometry. To conduct a stability analysis, the zero-jacobian condition (differential approach, Martinez [1983]) or the conjugate point condition (variational approach, Gillette and Dyson [19711) must be used. The limiting condition for stability, following the former method, is
/
371
Liquid bridge stahiliti’ data
that series. Although they might be approached asymptotically, their importance deserves a special analysis. These shapes are catenoidal, that is, portions of r = cosh z with the appropriate scaling. One of these equilibrium catenoids lies on the minimum volume limit and thus must be found
4~ ~2) ~K/aa aK/a~,
=
~A/8a 8A/~~
OK/8~,
8A/a~2
OV/0a 8V/0~
Table 1 Nondimensional minimum volume V. for liquid bridges at rest,
1I
=
0.
(6)
BV/0~5
For K and A fixed, the system formed by eqs. (3) and (6) consists of four equations with four unknowns, is a, chosen 4~,42 (the and equations V. which, ifare themultivalued), appropriate solution give the minimum stable volume, [ç~. A linear approximation [4] may be used advantageously for long bridges between slightly different discs: 19 ~ain
=
2A (2A
—
—
~
i
—
H J2/3 sin A/A ‘ (7)
2.2. Special cases All curves and graphs presented in this paper correspond to points that are singular in the transformation K, A, V~-sa, 4’~ ~2 according to eq. (6), but we refer now to singularities that remain in the computation after eq. (6) is accounted for, They correspond to the catenoidal stability limit (already mentioned in section 1 when mentioning the catenary as a special meridian curve), to the stability limit with zero slope (dr/dz = 0) at the larger disc, and to the stability limit of minimum Plateau undulation (minium a for all Vmjn at constant K; see table 1). Table 1 gives Vmjn versus A for various disc ratios K, and the internal parameters to compute the actual shape by eq. (2), as explained in appendix 3 2.2.1. Catenoidal limit For every disc ratio K, there is one series of shapes not dealt with in eq. (1) because it becomes singular for a = ir/2 that would correspond to
anchored to the edges of coaaial discs of radius
R1
and
R2.
as
a function of nondimensional disc separation \, and diameter ratio K; the parameters a and ~ serve to compute the interface shape (appendix 3): the liquid/solid angle at both ends. ~ and 9, is also presented to check for limiting contact angle conditions 0i
~
1.2504
1.3502 1.0526 0 8061 0.6018 0.4316 0.2882
24.7 41.0 540 64.3 72.4 78.8
24.7 41.0 540 64.3 72.4 78.8
5.984 7.118 8.372 9.915 11.855 14.270 17.245 20.871 25.247 30.477 36669 43.937
1.2075 1.1664 — 1.1002 —0.9770 —0.8346 —0.6620 —0.4253 0.2692 —t).5427 —0.6948 —0.8025 —0.8851
t).1660 0.0605 t).0000 0.0000 0.0000 0.0000 0.0000 1.5708 1.5708 1.5708 1.5708 1.5708
83.9 87.9 9t).0 90.0 90.0 90.0 90.0 90.0 90.0 90.0 90.0 90.0
83.9 87.9 90.0 90.0 9t).0 90.0 90.0 90.0 90.0 90.0 900 9t).t)
06 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2
1115 1.762 2.462 3.240 4.119 5.131 6.328 7.778 9.557
—1.5235 —1.4567 — 1.3970 — 1.3420 — 1.2885 — 1.2322 — 1.1689 — 1.0985 — 1.0263
1.3667 1.0903 0.8691 0.6993 0.5799 0.5124 0.4950 t).5188 0.5763
246 39.9 51.8 60.7 67.1 71.2 73.6 74.8 75.6
246 41.8 55.6 66.8 75.8 83.0 88.6 92.7 95.9
2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8
11.749 14.440 17.724 21.698 26.466 32.136 38.823 46.646
0.9608 —0.9129 —0.8920 —0.8990 —0.9260 0.9630 — 1.0029 — 1.0418
0.6649 0.7807 0.9113 1.0366 1.1420 1.2239 1.2857 1.3323
76.1 76.5 76.8 77.1 77.3 77.5 77.7 77.9
98.3 100.3 101.9 103.3 104.4 105.4 106.2 107.0
.1 K =
0.6 0.8 1.0 1.2 1.4 1.6
1
1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 K =
V
a
1.107 1.744 2431 3.187 4.025
—
1.5231 1.4567 13978 —1.3447 —1.2959
4,955
—
0.9
1. Martinez, .1. M. Perales / liqiod bridge stahl/ui’ ul~tui
372
Table 1 (continued)
Table 1 (continued)
S
.5
I’
K = ((.8 )).6 1.142 0.8 1.821 1.0 2.564 1.2 3.408 1.4 4.391 1.6 5.566 1.8 6.994 2.0 8.748 2.2 10.910 2.4 13.566 2.6 16.810 2.8 20.742 3.0 25.468 3.2 31.097 3.4 37.748 16 45 540 K = 0.7 (.6 1.194 ((.8 1.931 11) 2.752 1.2 3.701 1.4 4.834 1.6 6.212 1.8 7.906 2.0 9.996 2.2 12.57)) 2.4 15.723 2.6 19.556 2.8 31) 3 2 3.4
24.178 29.70)) 36.243 43.93))
A’ ().6 (1.6 1.278 (1.8 11) 1.2 1.4 1.6 1.8 2.0
2.108 3.043 4.142 5.470 7.11)1 9.114 11.600
o 1.5251 1.4568 — 1.3951 1.3365 —-1.2780 1.2178 — 1.1569 -—1.0988 1.0500 — 11)165 — 118)19 1.0054 -- 1.0224 --1.0474 - 11)760 —11052 — -
1.3921 1.1417 0.9510 0.8177 0.7383 0.7066 0.7147 0.7553 0.8220 0.9071 1.0001 11)901 1.1693 1.2348 1.2876 1 3295
24.4 38.5 49.3 57.1 62.4 66.)) 68.3 69.9 71.1 72.0 72.7 73.3 73.8 74.3 74.7 75 1
24.2 42.2 56.7 68.2 77.4 84.7 90.4 951) 98.6 101.6 104.1 106.1 107.9 11)9.5 110.8 112))
1.5277 — 1.4573 — 1.3932 1.3323 -- 1.2731 --1.2161 1.1645 --1.1223 -— 1.0936 — 1.0803 ~- 1.0811
1.4235 1.2019 1.1)411 0.9373 0.8836 1)8713 1)8924 0.9387 1.0018 11)725 1.1425
24.6 37.5 47.1 54.1 59.0 62.5 651) 66.8 68.3 69.4 7)).4
23.5 42.2 57.1 691) 78.5 861) 92.1 971) 101.)) 104.4 107.3
—
1.0928 1.1113 1.1335 1.1570
1.2059 1.26)32 1.3053 1.3422
71.2 71.9 72.5 731)
109.7 111 8 113.6 115.2
—
1.5307
1.4551
25.7
22.7
1.4581 --1.3926 -- 1.3317 --1.2757 — 1.2265 1.1872 ‘- 1.1605
1.2642 1.1314 11)517 11069 1.0177 1.0452 1.0901
37.1 45.8 52.3 56.9 60.4 62.9 651)
421) 57.4 69.6 79.4 87.3 93.8 99.1
—
—-
—
2.2 14.655 ‘ 1.1472 1.1436 66.6 103.5 2.4 18.384 -- 1.1461 1.1984 67.9 0(7.2 2.6 22.895 1.1543 1.2493 691) 11)1.4 2.8 28.307 1.1686 1.2941 69.9 113.1 31) 34.740 - 1.1864 1.3321 70.7 115.4 3.2 42.323 — 1.2056 1.3638 71.4 117.5 _____________________________________________________
I’
K = 0.5 0.6 1.408 0.8 2.374 1.)) 3.471 1.2 4.772 1.4 6.357 1.6 8.311 1.8 10.725 2.0 13.700 2.2 17.345 2.4 21.772 2.6 27.104 2.8 33.467 3.0 40.992
o
~i
.
1.5327 1.4591 1.3940 1.3364 1.2874 1.2493 1.2232 1.2098 1.21)73 1.2131 1.2244 1.2388 1.2547
1,4808 1.3233 1.2180 1.1601 1.1409 1.15)13 1.179)) 1.2177 1.2593 1.2993 1.3351 1.366)) 1.3921
28.) 38.)) 45.7 51.6 56.)) 59.3 62.)) 641) 65.7 67.1 68.2 69.2 7)1,1
22.1 42.0 57.8 70.5 80.7 89.)) 95.7 101.4 106.1 10).! 117.5 116.4 119.))
A = (1.4 1)) I 616 )).8 2.775
--
1 S114 1.4599
1 497)) 1.3765
3”— 4)).5
42.7
11) 1.2 1.4 1.6 1.8 21) 2.2 2.4 2.6 2.8
4.098 5.677 7.607 9.987 12.923 16.530 20.928 26.245 32.612 40.168
1.3989 — 1.3488 1.3107 — 1.2856 1.2719 — 1.2686 —1.2727 --1.2816 1.2933 1.3)163
1.3000 1.2631 1.2566 1.2702 1.2958 1.3254 1.355)) 1.3821 1.4059 1.4262
47.1 52.3 56.4 59.5 621) 641) 656 671) 68.2 69.2
58.9 72.)) 82.5 91.1 98.2 11(41) 11(91) 113.2 116.8 1193)
(1.6 ((.8 11) 1.2 1.4 1.6 1.8
1.976 3.405 5.048 7.019 9.430 12.398 16.047
1.5252 1.4617 1.4108 -~1.3732 1.3489 1.3362 — 1.3327
1.5057 1.4254 1.3786 1.3607 1.3628 1.3765 1.3953
38.7 451) 50.3 54.7 58.1 60.9 63.1
24.7 44.9 61.3 74.5 85.2 941) 101.3
21) 2.2 2.4 2.6
20.509 25.919 32.418 40.154
1.3357 1.3427 ‘— 1.352)) 1.3623
1.4151 1.4337 1.4502 1.4645
64.9 66.5 67.7 68.8
107.3 112.4 116.8 12)).5
1.517))
1.5155
47.3
29.9
1.471)3 1.4367 1.4157 1.4063 1.4025 1.4047 1.408) 1,4169 1.4244
1.4748 1.4548 1.4505 1.4544 1.4653 1.4762 1.4869 1.4965 1.5049
51.7 55.6 58.9 61.1 63.7 65.6 67.1 68.4 69.4
49.5 65.6 78.8 88.3 98.2 11(5.5 111.6 116.7 121.1
-
— . —
A = (1.3 —
—
.
A = ((.6
2.656
((.8 11) 1.2 1.4 1.6 1.8 21) 2.2 2.4
4.475 6.592 9.142 11.852 16.092 20.777 26.467 33.318 41.488
--
-‘ --
— —
I. Martinez. f.M. Perales
V
9-
a
4 105
—
1 5214
1 5386
58 7
39 9
0.8
6.536
—
1.5002
1.5279
61.1
1.0
9.416
-
1.2
12.910
1.4876 1.4819
1.5244 1.5252
63.4 65.4
58.3 73.5 86.0
1.4 1.6
17.182 22.400
1.8 2.0
28.737 36.365
1.4808 1.4825 1.4857
1.5284 1.5324 1.5363 1.5400 1.5432
67.2 68.6 69.9 70.9 71.8
96.2 104.6 111.6 117.4 122.3
2.2
45.462
373
Liquid bridge stability data
the general case in eq. (6), and is analyzed apart
Table 1 (continued) A
/
—
—
1.4897 —
1.4939
because the following peculiar property [11]:K) it splits theofminimum volume curve (for constant in two regions, such as for longer bridges the neck travels towards the larger disc during the breaking process, whereas it narrows without axial shifting
for shorter bridges. In this case ~,,
(8a)
=
— =
—arcsin(s/i
2 /sin a),
(8b)
K
—
=J(a) =0. separately (for every disc ratio K) as presented in table 2. This limiting case has already had direct application in the study of disjoining pressure effects on thin liquid films on a solid surface [14], due to its relevance as the most sensitive liquid bridge configurations (the pressure jump across the interface is zero). Exact knowledge of these shapes is required to evaluate their effect when very small forces show up [14], so that, although the computations in this case are not so involved as in the other cases, the explicit form of the determinant corresponding to eq. (6) is presented in appendix 2.
2,2.2. Stability limit with cylindrical ending (dr/dz =
K 1.0 0.9 0.8
results are presented in table 4.
when solving eq. (6), and the
Table 3 Minimum volume shape with cylindrical ending (dr/dz
0~ 12.2 11.7 11.4 11.3 11.2 11.2 10.8 9.8
K 1.0 0.9
a
1.1407 —1.1476
~i
0.0000 0.4984
A 2.1283 1.8620
0.8 0.7
—1.1618
0.7128
—
1.1825 1.2099
0.8813 1.0257
0.5071 0.5017 0.4747
12.2 13.0 14.3 16.2 19.0 22.8 27.9 34.6
—
1.2447
0.4118 0.2869
1.0878 0.7996
43.5 55.5
7.9 4.7
0.2 0.1
1.2880 — 1.3410 — 1.4049 — 1.4810
0.2 0.1
—0.9201 —0.6433
2.6721 3.1881
0.4718 0.4738 0.4801 0.4900 0.5007
=
0) at
the larger disc; for every disc ratio K, the separation A and the corresponding volume V~,, are given: with K, a and ~, the actual shape may be computed as explained in appendix 3. these points correspond to curve B in fig. 2 0i ~:
Oi
0.3
0.4
a
V 0.7129 0.7218 0.7517 0.8058 0.8842 0.9801 1.0761 1.1354
A
0.5
has been asymptotically approached numerically by reducing
2.2392 2.2823 2.3079 2.3172 2.3180 2.3250 2.3599 2.4552
0.6
2,2.3. Stability limit with minimum undulation As mentioned above, the equilibrium shape of an axisymmetric liquid bridge in zero gravity corresponds to a piece of an axially periodic curve (Plateau curve) that can be identified by the hollow/summit radii ratio (cos a), as shown in fig. 1. For every disc ratio K, there is a minimum volume stability limit with a minimum value of a; it
This is really a simplification with respect to
—2.2392 —2.1745 —2.0791 — 1.9501 — 1.7893 — 1.6015 — 1.3926 — 1.1658
0.7
Results are shown in table 3. For the special case of equal discs (K = 1), eq. (8c) reduces to 2E(a, IT/2) — F(a, ~r/2) = 0, which gives A = 2.128392, an inportant value that has appeared with different degrees of error in the literature.
0) at the larger disc
Table 2 Catenoidal stability limit; for every disc ratio K. a separation exists, such as the minimum stable volume V,,,~.,is a catenoidal shape stretching from z 1 to z, when scaled with the neck radius (that is, r = cosh z): 9~and 0, are the angles the liquid forms with the discs: these points correspond to curve C in fig. 2
(8c)
0.6 0.5 0.4 0.3
—
—
—
V
7.901 6.748
90.0 74.0
90.)) 90.0
1.7832
6.862
68.1
90.0
1.7270 1.6778
7.246 7.834
64.2 61.4
90.0 90.0
1.1535
1.6284
8.623
59.8
90.0
1.2677 1.3689 1.4561 1.5264
1.5726 1.5036 1.4111 1.2739
9.630 10.890 12.454 14.389
59.1 59.6 61.7 66.2
90.0 90.0 90.0 90.0
1.
374
1.’lartine:. .1. ¶.,J• Pi,u’o/u’~
liquid (‘ridge ta/ui/u ii’ ,/ata (a)
Table 4 Minimum solume corresponding to minimum undulation ( minimum): for ever\ A’ there is a minimum us ersus ‘I (see table 1) and it is presented here: these points correspond iii curse A in fig. 2
0.1
20
0.5
-,‘
-.
‘ ~‘ ~,‘
V ,
/
o ((.0000 (1.8913
(11(000 ((94(17
31416 S 2845!
I 739 19 18.556
90,0 ~i 76.9
6~ 901) 102.2
((.8 ((.7
1.01(12 11(790
1.0261 1 1028
2.656! 2.4851
17.839 17.265
72,9 69.9
104.7 !((5 7
((.6 (1.5 ((.4 (1.3 (12 ((.1
1.1452 I 207)) 1.2685 1.3327 1.4()25 1.4808
1.1763 1.2492 1.3221 13948 1.4650 1 5280
2.3185 2 1513 1.9778 I 7949 1.5945 1.381)1
16,777 16.389 16.1)93 15.944 15,975 16.717
67.4 65 3 63 8 63! 63 7 67.0
105.8 1(15,)) 103.4 101.1 9S1)
‘/
// ‘
/
10
~/
-
/
/‘
=
‘~
1
—~
‘ ,
/
-
,f
‘‘
_____________________________________________________________________
A’ 1)) ((.9
K
‘
/
1
/
/
~
/
/
‘
/
~
N/s
/
/
‘
/ //~/
5
/
/
,~
/
.
,
~)
/
/
/5
/~‘
0I
NT
0
3. Results and conclusions
1
2
A
3
(b) 20
The minimum volume for stability, versus bridge length, is presented in fig. 2 for several disc ratios K. The special cases described above have
S
‘
\
\~
‘
\
S
,
\
~
31
~-‘-~ ‘
S
.
28
V
been explicitely included. High precision values can be found in the tables, The method to conipute the limiting shapes for the points included, as well as how to extend the tables here given, are As an example, let us consider the bridges that presented in appendix 3. can be formed between discs of R 1 10 mm and R, = 20 mm in diameter (R1/R, = 0.5). assuming a certain liquid having some 30° contact angle with the solid discs in the presence of the outer fluid. A short liquid bridge of L — 9 mm in length (A =0.6) can 3.6) easily established by putting V= If 3 (U— of he liquid in between (fig. 3a). 1.5 cm the liquid is now gently withdraw at constant disc
\\\\
~\‘s\\’
S
.
‘
\ss”
“
\
‘~ \,
1
“ N N
~‘
10
~“=
-
N
~‘
~
\*
N,,
N
~
~
‘—id~-..
-.-
—.
=
~
C
0.1 =<_—~-
0
-
-s~ ‘
=
~-
— --—~—~
0.5
.
______ K
separation. the liquid border will detach from the
1’i~.2. Minimum volume F
disc edge when the angle at the edge of the larger disc diminishes to 30° (fig. 3h). well before the minimum volume V — 1 .4 given in table 1 is reached (fig. 3c). Assume now that one manages to have a long 3 liquid bridge 30 mm = 2) containing 9 cm of liquid (V=of21.3. fig.(A3d); when will such a
euiavial discs of radii R1 and R2 a distance / apart. 1 sr every A’ ‘— R1 R there are three special points in the I, ( .5 curve: )A( bridge of minimum undulation: (B) bridge of dr d:
for a liquid bridge 5inuliored
bridge break if stretched at constant volume’! (fig. 3f). or. alternatively, when will it break if liquid is removed at constant disc separation? (fig. 3e). Answers to these and similar questions are now
o
(cylindrical ending) at the larger disc: ((‘1 caienuuidal
bridge (a)
1) for constant A: )h) ~
(K (for constant
easily obtained from table 1 (or fig. 2). For K = 0.5 and V= 21.3 we see in table 1 that ~ —2.4 (36 mm long), whereas for K = 0.5 and .1 — 2 the minimum volume is V — 13.7 (5.8 cm3). In both cases. 0, being greater than 90°, the breaking
I. %!arone:, f.M. Peru/es
/
375
I.iquud bridge stability data
In conclusion, this paper presents (as far as we
know for the first time) a comprehensive and highly compact data set on the minimum stable
volume for liquid bridges between non-equal discs in absence of gravity; and not only the value of ---I
actual Vmin ~5shape gi~’~’~, of I~~1t the data freenecessary surface in to this compute limiting the configuration is provided.
~ -
Appendix I. Shape computation for given .
K,
A
and
V
Several methods have been investigated to compute the shape from primary data K, A
~
-
-
N
I Fig Fxample of liquid bridge shapes for Ri — 10 mm, R, —‘ 2)) mm (a) Stable bridge with .S — 1)6 and V= 3.6. computed with eq. (9). (h) While removing liquid from the (a) shape, the liquid would recede form the disc edge if the liquid/solid limiting contact angle is assumed to he 30° (see section 3). (c) Minimum volume stability limit for K = 0.5 and A = 0.6. computed as explained in appendix 3. (d) Stable bridge with A = 2 and V= 21 3, computed with eq. (9). (e) Minimum volume stability limit for K 0.5 and A = 2. computed as explained in appendix 3. (f) Stability limit when stretching from (d) at constant volume,
and V. In
all cases, the problem being highly nonlinear, a fairly good initial guess is required; eq. (5) is an If an algebraic solution of eq. (3) is desired, we
evaluable help in that respect. recommend the specification of a and computation ~ and ~2 from the first two conditions in eq. (3); the first one may be solved explicitly as /, = ~,(K, a, ~) and the second one A(a, ~ 4~)= A by a standard root finder, thus obtaining a V = V(a) relationship. Another approach is by a series expansion in terms of orthogonal polynomials, for instance Tschebyscheff economization series. Thence, using L/2 as unit length, the shape r = r( z) can be
written as N
shape (readily obtained from have a hollow and a summit.
K,
a and 4~)will
As regarding the relevance of this results to the real floating zone problem, it might be argued that
r(z)
n=0
with theNnth order Tschebyscheff polynomial, where7 the ± 1 coefficients a, 1 are to be found from the conditions r( —1) = R1/L, r( + 1) =
true molten zones do not have fixed volume or length as the model assumes, but the mchanical stability of a real floating zone (be it a melt or a solution) to have much smaller characteristic happens time scale that athermal or physico-
R2/L.
chemical instabilities and thus, for small time intervals, it is at all licit to treat the liquid volume and length as constants, as has already been suc-
and
cessfully made in [15], where this same liquidbridge model has been used to predict, with high
accuracy, the outer-shape evolution (stable and unstable) of a real floating zone of molten silicon aboard an orbiting laboratory.
a,
=
2 (1 r
f
dz
v =
—
1
r~/1+ (dr/dz)2 dz
=
minimum
for every possible variation of the coefficients. If the series is truncated at N = 3, a 0, a1 and a2 can be explicitly solved for and the minimum of the
376
I. .i’larri,iu’:, .1. .%/. Peso/eu
l.iquid bra/ge siahuliti’ datut
last integral with a~ easily found. Unfortunately this expedient procedure gives an accuracy of the same order as the straightforward expression of eq. (5). As another approach, a very accurate solution can be found by direct integration of the differential equations for constant mean curvature (Laplace capillary equation) in the form
•t’(
dr
i’(:~)=0,
(l3h)
fv(:) cosh : d: =0.
(l3c)
.
with
rI, 1 =1—H --
dt
I +
~2
‘
—~(l +
,.
=
t-)
‘
with
(Jacobi condition in the variational formulation) is 2i’ d t’ d ~ —2 tanh :-~- +p cosft: =0. (12) where p is an internal constant that, in addition to the two boundary conditions in eq. (12), is to he found from the three conditions
‘I~
(9a)
/
=
t~.
)
.~
-
-
which, upon substitution of the general solution of eq. (12) takes the form
9h )
dt’
with
~r
=
“I
=
sinh
where the internal variable t (the local slope of the shape) and the nondimensional pressure jump aceross the free surface p (constant) are introduced. In this scheme H, H = (1 — K)/(1 + K), is given, and p and t~ are estimated with the aid of eq. (5). A standard Runge—Kutta numerical integration of eq. (9) from : = 0 to : = 2A (.‘~ is given) then yields new values for H and V =
r
I
V = (1
~
—1.
(lOa)
8’ I =~ —
H
+
r
)3
.1/
0. (9c)
H
( l3a)
0.
=
(lOb)
and a standard Newton—Raphson iteration is used to approach the specified H and V to the desired accuracy. The procedure, of course, becomes singular near the stability limit. Appendix 2. Stabilit~ limit corresponding to a catenoid Quasi-catenoidal shapes can be expressed. when properly scaled, as
(cosh 2:,
I)
sinli cosli
uosh’1
siili coshi
costs’:,
/) :~(
ZS(
g(
:~)
g)
I
cosh 2:~(/4 ‘
where the functions -
f(:)
=
—
g(:)
=
~-
f(:) and
.
sinh(2:)
—
+
+ ~ sinh(2:) + .
g(:) are
(14)
-
~-
.~,
cosh(2: ). sinh(4:).
(ISa) (15h)
‘
After substitution in eqs. (14) and (15) ol cosh :, = arg cosh K ,
(16)
the only remaining unknown is Zi. which is to he found from a nontrivial solution of eq. (14). After is found the nondimensional bridge length and liquid volume are given by 2 — t , (l7a) cosh :, + cosh sinh(2z, ) + 2:~ — sinh(2: ) — 2: V=2 (17h) (cosh :2 + cosh :~) These values can he found in table 2 as a funtion =
of disc ratio K. Appendix 3. Summary of computational procedures
r(:)=cosh:+s’(:).
(11)
where is a small number. After substitution in the equilibrium equation (Laplace capillary equalion, or Euler—Lagrange equation in the variational formulation) the resulting stability equation
How
to find
the shape for points in tab/es’ 1. .1 and 4
(1) Given K. a and ~. (2) Conipute ~ = ~( K, a, /~) from eq. (3a): function ~, below.
I. Martinez, f.M. Peru/es
(3) Compute the shape z = z(a, ~, ~ ~, r= r(a, ~, 4~,~); PROCEDURE Shape below.
/
377
Liquid bridge stability data
How to find new points
(3) Interpolate a(K, A) from table 1. (4) Compute 4~= ~ 1(K, A, a) from eq. (3b); function 4~below. (5) Compute Vmjn with eqs. (3c) and (4) and the
(1) Given K and A. (2) Compute AB(K) from eq. (7) or points in curve B, fig. 2.
shape (z, r) as above. (6) To check or enhance accuracy, evaluate the jacobian in eq. (6), slightly change a, and repeat steps 4 to 6.
Pseudocode for the computer routines mentioned above
FUNCTION r
=
r(cL,~)
(1_sin2asin2~)~2
FUNCTION ~2(K, a, 2 =
~
±arc sin VI1~r(d1,~1)/K) sin a
use + if
A
-
if A>AB
PROCEDURE Shape(a,~ 1,42,z,r,n)
! z & r are vectors of size n
for i0 to n ~~+( qa,- q~)(i/n) P1ateau~cs,i~,z(i),r(i)) next i
FUNCTION ~1(K,A,cs) =
always 0<~
arc sin S ~flO
loop ~2=qi2(K,
a,
~)
Plateau(cs, P1,z1,r1) Plateau(a, 1~9,z2,r2) NewA’~(z2-z1)/(r2+r1) repeat ioop, changing ~ until NewAA
PROCEDURE Plateau(cs, ~,z,r) Elliptic integrals(ct,~,F,E) zcosruF+E rr( a, qi)
! standard library routine
1. .3-lartine:. .1. 1.1. Pera/e,v
378
,/
Luquid bridge sw/u/dr i/ala
IXI
References
L.A. B&uueher and M.J.B. Evana J. (‘olloid Interface Sc). 75 (1980) 409.
I~IR.A.
III
RD. (rillette and l’).C 196
121
SR. (one]) and (1977) 466.
131
A. Sanz and I. Martinez, J. (‘olloid (1983) 235
141
I. Martinez. in: Material Sciences under Microgravity. ESA SP-191 (ESA, Paris. 1983) PP 267—273
1131
151
J. Mcsegucr, J. (‘rystal Growth 67 (1984) 141. J. Mcseguer. in: Material Sciences cinder Microgravitv, ESA SP-222 (ESA, Paris. 1984) pp 297- 30)(
under Microgravitv. ESA SP-222 (ESA. Paris. 1984) pp. 173-- 181. 1141 J.F. Paddav, in: Material Sciences cinder Mucrogras’it\.
I (21
I~I SR.
Dyson, (‘hem. Eng. J .3 (1972)
MR. (‘ordcs, J. Crystal Growth 42 Interface Sc). 93
Coriell. S.C. Hardy and MR. Cordes..l. Interface Sc) 60 (1977)126
Brown and L.L. Scruvcn, Phil. Trans. Roy. Soc. london A297 (1980) 51
1101 1.1-1. l.Jngar and R.A. Brown, Phil. London A306 (1982) 347.
Trans. Roy. Soc.
Ill J. Meseguer and A. Sanz. J. Fluid Mccli. 153 (1985) 83. {12I A. Eyer. B.O. Kolhesen and R. Nitsche. 3. Crystal Growth
(‘olloid
1151
~7 (1982)145. A. Ever. H. Leiste and R. Nitsehc, in: Material Sciences
ESA SP222 (ESA, Paris, 1984) pp. 9-14. I. Martinez and A. Eyer, J. Crystal (irowth 750986) 535.