Langmuir 2004, 20, 7659-7664
7659
Line Energy and the Relation between Advancing, Receding, and Young Contact Angles Rafael Tadmor† Department of Chemical Engineering, Lamar University, Beaumont, Texas 77710 Received March 7, 2004. In Final Form: June 1, 2004 The line energy associated with the triple phase contact line is a function of local surface defects (chemical and topographical); however, it can still be calculated from the advancing and receding contact angles to which those defects give rise. In this study an expression for the line energy associated with the triple phase contact line is developed. The expression relates the line energy to the drop volume, the interfacial energies, and the actual contact angle (be it advancing, receding, or in between). From the expression we can back calculate the equilibrium Young contact angle, θ0, as a function of the maximal advancing, θA, and minimal receding, θR, contact angles. To keep a certain maximal hysteresis between advancing and receding angles, different line energies are required depending on the three interfacial energies and the drop’s volume V. We learn from the obtained expressions that the hysteresis is determined by some dimensionless parameter, K , which is some normalized line energy. The value of K required to keep a constant hysteresis (θA - θR) rises to infinity as we get closer to θ0 ) 90°.
Introduction The shape of a drop resting on a surface depends on the material properties of the drop, the air (or vapor) around it, and the surface on which it is placed. This is usually described as a function of the interfacial tensions by the Young equation1
γSL + γ cos θ0 ) γSV
(1)
where γSL, γ, and γSV are the interfacial tensions between the liquid and the solid, the liquid and the vapor, and the solid and the vapor, respectively, and θ0 is the equilibrium contact angle the drop makes with the surface. To derive the Young equation, normally the interfacial tensions are described as forces per unit length and from the onedimensional force balance along the x axis (see Figure 1a), eq 1 is obtained. From eq 1 it appears that there is only one thermodynamic contact angle (θ0); however, our daily experience shows that drops have a spectrum of contact angles ranging from the so-called advancing contact angle, θA, up to the so-called receding contact angle, θR, which are the maximal and minimal values the contact angles can obtain. To understand the reason for the existence of a spectrum of contact angles, we need to consider, apart from the surface tensions, also the three-phase contact line which gives rise to yet another term that contributes to the total energy of the system. The line energy associated with the three-phase contact line was a subject of many studies.2-5 From these studies it was established that the line energy is a result of defects or imperfections in the smoothness (either chemical or structural) of the solid surface. This †
E-mail:
[email protected].
(1) Young, T. An essay on the cohesion of fluids. Philos. Trans. R. Soc. London 1805, 95, 65. (2) Israelachvili, J. N. Intermolecular & surface forces, 2nd ed.; Academic Press Limited: London, 1991. (3) de Gennes, P. G. WettingsStatics and Dynamics. Rev. Mod. Phys. 1985, 57 (3), 827-863. (4) de Gennes, P. G.; Brochard-Wyart, F.; Quere, D. Capillarity and Wetting Phenomena; Springer: New York, 2003. (5) Adamson, A. W.; Gast, A. P. Physical Chemistry of Surfaces, 6th ed.; John Wiley & Sons: New York, 1997.
is a very important conclusion as it shows that the oversimplistic thought of the contact line as giving rise to a “line tension”sas if it is some kind of belt or string that presses on the dropsis incorrect.6 Rather, the line energy is related to local blemishes on the surface and its value is associated with the local type and concentration of those blemishes. Being of local nature, one can in principle have different values for the line energy at different regions of the line if the concentration or type of blemishes is different in those regions. Moreover, these blemishes act to pin the contact line, and it is this pinning effect that gives rise to angle hysteresis. Hence the same blemish specifications may pin it in an advancing direction or in a receding direction depending (for example) on the way the surface supporting the drop is tilted. Thus, the surface defects determine the maximal absolute values of the contact angle deviation from the equilibrium Young angle (maximal advancing, θA, and minimal receding, θR, contact angles). The defects only determine the values; however the absolute values of angular deviations-from-equilibrium are different, |θA - θ0| * |θR - θ0|, as we show later. If we consider a completely isotropic system, then the maximal interaction associated with the maximal advancing angle should equal the negative of the interaction associated with a minimal receding angle. Hence the actual contribution of the line energy is different for the same drop and the same surface (with the same defects) depending on the local contact angle (e.g., the opposite sign described in the previous sentence). In this study, rather than relating the line energy to the surface imperfections that caused it, we relate it to the contact angle that resulted from these surface imperfections (and the other parameters in the system). To see the contribution of the three-phase contact line to the energy, we need to describe the problem in its correct three-dimensional perspective (Figure 1b) rather than the one-dimensional approach of Figure 1a. We first derive the Young equation (eq 1) using a three-dimensional geometry, ignoring the three-phase contact line, and see (6) de Gennes, P. G. Two remarks on wetting and emulsions. Colloids Surf., A 2001, 186 (1-2), 7-10.
10.1021/la049410h CCC: $27.50 © 2004 American Chemical Society Published on Web 07/30/2004
7660
Langmuir, Vol. 20, No. 18, 2004
Tadmor
Figure 1. (a) Two-dimensional representation of a drop on a surface describing interfacial tensions as forces balanced along the x axis which results in eq 1. For this, the contact line is viewed as a point object on which the force balance is made. (b) Threedimensional representation of a drop on a surface. Here the drop is three-dimensional, and the surface tensions can be viewed as surface energies. Then we can obtain the Young equation from surface energy minimization.
that indeed in this case there is only one contact angle. Then we incorporate the contact line into the equation and see how it can account for a spectrum of contact angles. The Young Equation Obtained with a Three-Dimensional Description The surface tension can also be viewed as surface energy (energy per unit area of the surface). Equation 1 can then be obtained by energy minimization2 as described below. Consider a drop floating in space (as if before it is placed on the surface). In this case it has a shape of a perfect sphere of radius RS (in the air or the medium around it), which is also the smallest surface-to-volume ratio for that drop volume. At this stage the surface tension of the drop, γ, corresponds to the pressure difference, ∆P, between the inside and outside of the drop according to the Laplace7 relation: ∆P ) 2γ/RS. Now suppose we bring a solid flat surface close to that drop, so that they can touch at a point. As the drop touches the surface at a point, it may spread to more than just that point depending on all the interfacial tensions (interfacial energies) in the system. These include the surface-drop interfacial tension, γSL, the surface-vapor interfacial tension, γSV, and the dropvapor interfacial tension γ (see Figure 1b for the description of the corresponding surfaces in the problem). In this case the drop will adopt a shape of a cap that is part of a sphere with radius, R, which is bigger than RS and thus reduce its internal Laplace pressure (because R > RS). As the drop covers more surface area, the cap shape it adopts corresponds to an ever increasing sphere size and lower pressure. During this spreading process, the flat area between the drop and the surface, ASL, is ever increasing, and the exposed-to-the-air substrate surface flat area, ASV, is ever decreasing. On the contrary, the change in the cap area of the liquid vapor interface, ALV, is not monotonic, and the equation for it is given in Appendix 1 (eq A1.8). ALV increases or decreases depending on the angle the cap makes with the surface: at higher contact angles it decreases with the decrease in contact angle, θ, and at lower contact angles it increases with the decrease in contact angle (changing direction at θ ) 90°). At a certain point, however, the drop will stop advancing, and reach equilibrium, because the net reduction in surface energy becomes zero. In other words, if we consider the total amount of work change, δw, as a result of this spreading process, we can write
δw ) γ dALV + γSL dASL + γSV dASV + ∆PdV (2) where dV is the change in volume as a result of the change in the Laplace pressure. When δw ) 0 the spreading will (7) Laplace, S. d. Mechanique Celeste, Supplement to Book 10; 1806; Chapter 1, p 419.
stop, and a stable drop will form on the surface. Before we rewrite eq 2 for δw ) 0, note also that dASL ) -dASV since ASL (the flat contact area of the drop and the surface) is increasing at the expense of ASV (the flat contact area between the surface and the air)ssee Figure 1b. We therefore write them both in terms of ASL: dASL ) -dASV; also note that all the changes in the volume, dV, are a result of the density changes in the liquid drop, but since liquids are almost incompressible, we can neglect dV with respect to the rather large changes in dASL and dALV. (This assumption may be oversimplified in some cases when the density changes of the liquid are large.) Thus at equilibrium δw ) 0 and we get
0 ) γ dALV + (γSL - γSV) dASL
(3)
and after dividing both sides of eq 3 by dASL we get
γ
dALV + γSL - γSV ) 0 dASL
(4)
Note that the derivative dALV/dASL should be done at constant volume. In Appendix 1 we show that at constant volume
dALV/dASL ) cos θ0
(5)
and thus eq 4 becomes the familiar Young equation (eq 1)
γ cos θ0 + γSL - γSV ) 0 Incorporating the Contact Line into the Young Equation As noted earlier, to obtain a spectrum of contact angles, θR < θ < θA (rather than the single thermodynamic equilibrium θ0 according to eq 1), we need to introduce the contact line as an extra term in our energy minimization. Thus we rewrite eq 3 together with the energy term for the three-phase contract line
δw ) 0 ) γ dALV + (γSL - γSV) dASL - k dL
(6)
where L is the length of the three-phase contact line (circumference of the drop substrate contact area) and k is the energy per unit length associated with a length increment of a contact line dL. The term k dL was given an arbitrary minus sign in eq 6 to stress the fact that the contact line tends to pin to the solid surface, and thus the line energy contribution always opposes the contributions of the other terms, i.e., opposes the progress toward an equilibrium Young contact angle. Thus advancing and receding contact angles correspond to opposite directions of the line force with respect to the three-phase line
Line Energy and Relation between Contact Angles
Langmuir, Vol. 20, No. 18, 2004 7661
position. As before we divide both sides of the equation by dASL and obtain
dALV dL + γSL - γSV - k )0 dASL dASL
γ
(7)
Note that both dL, dASL, and dALV are subjected to constant drop volume: V ) constant (the drop may alter its shape but not its volume). In Appendix 2 we show that for constant volume
(
)
3 π (2 - 3 cos θ + cos θ) dL ) dASL 3 V sin3 θ
1/3
(8)
and since
dALV/dASL ) cos θ we get
(9)
(
)
3 π (2 - 3 cos θ + cos θ) γ cos θ + γSL - γSV - k 3 V sin3 θ
1/3
)0 (10)
Note that we shifted from θ0 to θ. This is because, as we shall see, now θ is different from the equilibrium Young contact angle. Equation 9 is different than eq 5 because the angles are different. The ratio of the surfaces dALV/ dASL remains cos θ although θ differs from the Young contact angle, because it is purely a geometrical relation (see Appendix 1). To see how θ can vary, we write eq 10 so as to find out how k can be expressed as a function of θ, that is how much line energy we need to invest to get θ
(
k ) (γSL - γSV + γ cos θ)
)
1/3 V sin3 θ 3 π (2 - 3 cos θ + cos3 θ) (11)
For simplicity let us note the difference between the interfacial tensions of the solid with respect to the liquid and of the solid with respect to the vapor as ∆γ so: ∆γ ) -(γSL - γSV). Now we can write eq 11 as
(
)(
)
1/3 γ 3 sin3 θ cos θ - 1 ∆γ π (2 - 3 cos θ + cos3 θ) (12)
k ) ∆γV 1/3
From eq 12 we see that θ ) θ0 for the classical case where ∆γ/γ ) cos θ, and then k ) 0. We know, however, that the same drop can have different contact angles with the surface depending on the way it was placed (advancing or receding). For example if we tilt the surface a little, then different parts of the same drop will have different contact angles and hence different k contributions. To shed more light on the way k changes with the angle, let us rewrite eq 12 using the identity from the Young equation, ∆γ/γ ) cos θ0:
(
k ) ∆γV 1/3
)
1 cos θ - 1 × cos θ0
(
)
3 sin3 θ π (2 - 3 cos θ + cos3 θ)
1/3
(13)
From eq 13 we can learn that at θ ) 90°, k obtains the value k ) -(3V/2π)1/3∆γ, or, for units in which V ) 1 and ∆γ ) 1, simply -(3/2π)1/3 = -0.78.
Figure 2. K dependence on the contact angle, θ, for Young equilibrium contact angles, θ0, smaller than 90°. The five different lines represent five different Young equilibrium contact angles θ0: from left to right the Young contact angles correspond to 1/cos(θ0) equals 1.01, 1.1, 2, 20, and 107 which roughly correspond to θ0 equals 8°, 25°, 60°, 87°, and 89.99999°, respectively. The line corresponding to θ0 ) 89.99999° is dashed and for the scale of the figure it coincides with a vertical line along θ0 ) 90°. Note that the slope at which K intersects the x axis increases as θ0 approaches 90°.
We can also learn from eq 13 that k is proportional to the cube root of the drop’s volume. Since k is a function of surface irregularities (defects on the surface), then a fixed condition of surface irregularities fixes the maximal value of k as well. This maximal k value should correspond to the maximal advancing contact angle, θA. The negative of that maximal k should correspond to the minimal receding contact angle, θR. Hence the maximal hysteresis in angle ∆θ ) θA - θR will be smaller for drops of greater volumes so long as gravitation is negligible and the drop maintains its spherical cap shape. To compare the way k varies for different θ0, let us assume a variable K which will be identical to k for units in which V ) 1 and ∆γ ) 1. Thus for such units, eq 13 can be written as
K )
(
)
1 k ) cos θ - 1 × 1/3 cos θ0 ∆γV sin3 θ 3 π (2 - 3 cos θ + cos3 θ)
(
)
1/3
(14)
The definition of the variable K is important for the calculation of the Young equilibrium contact angle as described in the next section. Before we do that, however, let us plot K for different values of θ and θ0 and learn about its role in angle hysteresis. In Figure 2 we show K as a function of the angle that the drop makes with the surface for a few Young equilibrium angles below π/2. As implied from eq 14 all lines should meet at K ) 0 when θ ) π. At θ ) π/2 again all lines meet as noted earlier. An important phenomenon can be seen in Figure 2: the slope of K at θ ) θ0 (when K ) 0) increases as θ0 approaches 90°. Higher derivatives of dK /dθ at θ ) θ0 imply that more energy is required for the same deviation from θ0 as θ0 approaches 90°. In other words, for θ0 closer to a straight angle, more energy per length is required to deviate from θ0. As noted earlier, the energy associated with k is obtained from defects on the surface. Thus for
7662
Langmuir, Vol. 20, No. 18, 2004
Tadmor
Finding the Young Thermodynamic Contact Angle from Advancing/Receding Data
Figure 3. K dependence on the contact angle, θ, for four different Young equilibrium contact angles, θ0, at two angular intervals away from straight angle: θ0 ) 90° ( 30° and θ0 ) 90° ( 0.00001°. The two θ0 values which are closer to 90° cannot be resolved on the scale of this figure and are both represented as an overlapping dashed line (the vertical dashed line along θ ) π/2). On a larger scale they would look qualitatively as the other pair shown.
similar surfaces (with similar defects) k would be fixed and hence the higher slope of k implies that ∆θ ) θA - θR would be smaller for Young angles, θ0, closer to 90°. Indeed experimental results suggest that this may be the case (cf. Figure 4).8 For Young angles higher than 90°, the trend of increasing dK /dθ at θ ) θ0 with the approach of θ0 to 90° remains, and it is possible to plot a figure which will be somewhat parallel to Figure 2 for the case of θ0 > 90°. Rather than plotting such parallel figure, we show in Figure 3 four representative Young contact angles which form two couples, each of equal distance from θ0 ) 90° but from opposite directions. Two of the θ0 are 30° away from 90° (60° and 120°), and two are 0.00001° away from 90° (89.99999° and 90.00001°). We see that the two angles which are close to 90° practically coincide (dashed vertical line in Figure 3), and it is impossible to resolve them within the resolution of the figure. Had they been plotted on a larger scale, they would look qualitatively like the other pair (60° and 120°). The two θ0 angles farther from 90° demonstrate the fact that they are not mirror images of each other. The higher (120°) angle has a much lower maximal K absolute value than the corresponding minimal K absolute value for 60° (both can be readily obtained by solving dK /dθ ) 0, but the expressions are lengthy). Generally, the minimal k value for θ0 angles lower than 90° must be below -(3V/2π)1/3∆γ (below K ) -(3/2π)1/3 = -0.78), even if the θ0 is very far from 90° (approaches 0). On the other hand, the maximal k value for θ0 angles higher than 90° would decrease with increasing θ0 and would approach 0 when θ0 approaches 180°. This agrees with our experience as we observe that drops of very high contact angle can often roll off a surface easily while smaller contact angles often are less likely to slide down a surface. (8) Lam, C. N. C.; Wu, R.; Li, D.; Hair, M. L.; Neumann, A. W. Study of the advancing and receding contact anlge: liquid sorption as a cause of contact angle hyateresis. Adv. Colloid Interface Sci. 2002, 96, 169.
Perhaps a more important outcome which results from the introduction of the variable K is for finding the true Young “thermodynamic equilibrium” contact angle. For the very same drop resting on the very same surface, V, ∆γ, and θ0 are fixed, and the maximal absolute value that K can obtain is fixed as wellsit is only a function of the irregularities on the surface. If we further assume that the irregularities on the surface are isotropic in nature and in distribution, then the absolute value of K would be the same whether it is associated with a receding or an advancing contact angle. Hence the resistance to the motion out for an advancing drop will equal the resistance to the motion in of a receding drop because both of these resistances are a result of the pinning of the contact line to the similar blemishes. They would, though, be opposite in direction, i.e., opposite in sign. Thus eq 14 written for advancing contact angle should equal the negative of eq 14 written for receding contact angle
K (advancing contact angle) ) -K (receding contact angle) ) cos θA sin3 θA -1 cos θ0 (2 - 3 cos θ + cos3 θ )
(
-
)(
(
)(
A
A
)
1/3
)
cos θR sin3 θR -1 cos θ0 (2 - 3 cos θR + cos3 θR)
)
1/3
(15)
where the factor (3/π)1/3 was omitted from both sides of the equation. From eq 15, the equilibrium Young contact angle, θ0, can be readily calculated. By rearranging eq 15, we can write θ0 explicitly as
(
θ0 ) arccos where
)
ΓA cos θA + ΓR cos θR ΓA + ΓR
ΓR ≡
(
sin3 θR
ΓA ≡
(
sin3 θA
and
(2 - 3 cos θR + cos3 θR)
(2 - 3 cos θA + cos3 θA)
(16)
)
1/3
)
1/3
Experimental Evidence A careful study of contact angle hysteresis, ∆θ (namely, ∆θ ) θA - θR), using similar surfaces but different drops was done by Lam et al.8 The authors used a systematic series of hydrocarbon chains and observed that the advancing and receding contact angles become very close when extrapolated to 90° in the series. Using eq 16, we calculated θ0 from the different pairs of experimental θA and θR given in the paper by Lam et al. and then plotted the advancing and receding contact angles as a function of θ0 in Figure 4. We added to the figure the theoretical prediction of θA and θR according to eq 15 using K ) (0.128 (“-” for advancing and “+” for receding contact angles). From the figure we can see that the experimental results agree with the trend which eq 14 suggests, namely, the increase in θ0 toward 90° will decrease ∆θ for the same K . Note that the absolute value of K is constant only for the advancing and receding angles of the very
Line Energy and Relation between Contact Angles
Langmuir, Vol. 20, No. 18, 2004 7663
Figure 5. A schematic cross section of a drop on a surface. Neglecting gravity, it is exactly a part of a sphere of radius R.
Hence the drop is part of a sphere whose radius, R, we want to find. The volume of the drop, V, is a volume of a spherical cap. A volume of a spherical cap is a volume of an ice-cream cone (VI-C) minus the corresponding volume of the flat ending cone (VFC): The volume of the ice-cream cone is the triple integral
VI-C ) Figure 4. Advancing and receding contact angles as a function of the Young equilibrium contact angle. The points represent experimental data taken from Lam et al.8 The solid lines are the theoretical prediction according to eq 15 using K ) (0.128 (“-” for advancing and “+” for receding contact angles).
same drop. By assuming a constant K also for the different drops in the study of Lam et al., we should produce some error. However the agreement of the experimental results to the constant K simplification (Figure 4) shows that in that system K is primarily a function of θ0. In summary, we are able to calculate the Young equilibrium contact angle from the extrimal advancing and receding contact angles using eq 16. We can calculate the value of the line energy from the contact angle (be it advancing or receding), drop volume, and the interfacial tensions in the problem using eq 13 (or 12). We realized that the closer the Young equilibrium contact angle is to 90°, the more line energy is required for the same difference between θA and θR. We define a normalized line energy, K, whose value becomes infinite when θ0 ) 90°. Hence for a roughly constant K, the difference between the advancing and receding angles should diminish as θ0 approaches 90° and should becomes zero at θ0 ) 90° for any finite K . Acknowledgment. I thank M. Tadmor for useful discussions and T. Mahavier for going over some of the mathematical derivations. Appendix 1: Calculating dALV/dASL (≡dAC/dAP) with the Constrain of Constant Volume As this Appendix is purely geometrical, we denote ALV as AC so as to remember that this is the cap surface of the drop, and dASL as dAP so as to remember that this is the planar surface of the drop. Hence dALV ≡ dAC; dASL ≡ dAP; and dALV/dASL ≡ dAC/dAP. Another generalization that we make is write θ instead of θ0 becausesas it is a geometrical observationsit is not relevant if we are in the equilibrium Young angle or not. Consider the drop whose cross section is drawn in Figure 5. To lower its surface energy, the drop adopts a shape of a spherical cap. Neglecting gravitation, this spherical cap can be viewed as a part of a larger sphere of radius R, which is cut by a plane (the surface) at a certain place along the sphere. The intersection of the surface and the drop creates a circle of radius r ) R sin θ. To determine the exact angle θ, we need to know what is the size of sphere of radius R that the cap is taken from.
2π θ R r 2 sin R dφ dR dr ) ∫φ)0 ∫R)0 ∫r)0
2 πR 3(1 - cos θ) (A1.1) 3 The corresponding flat ending cone’s volume is its base area, πr2, times its height/3 (h/3)
VFC )
1 2 πr h 3
(A1.2)
As the flat ending cone and the ice-cream cone are both taken from the same sphere of radius R, we can use the geometrical relations
r ) R sin θ
and
h ) R cos θ
and rewrite eq A1.2 as
VFC )
1 1 πR 2 sin2 θ R cos θ ) πR 3 sin2 θ cos θ 3 3 (A1.3)
Hence the volume, V, of the spherical cap that we are interested in is
V ) VI-C - VFC ) 1 2 πR 3(1 - cos θ) - πR 3 sin2 θ cos θ ) 3 3 π 3 R [2 - 3 cos θ + cos3 θ] (A1.4) 3 Hence the radius can be expressed in terms of the volume, V, and the angle θ
[
π R ) V 1/3 (2 - 3 cos θ + cos3 θ) 3
-1/3
]
(A1.5)
Now we need to find expressions for the area of the cap interface AC and the area of the planar interface AP. The last is the circle area: AP ) πr2 and using the relation r ) R sin θ we get AP ) πR2 sin2 θ, which using eq A1.5 results in
[
π AP ) πV 2/3 (2 - 3 cos θ + cos3 θ) 3
-2/3
]
sin2 θ (A1.6)
The area of the spherical cap: The element of angle dθ corresponds to an element of arc R dθ. Multiplying it by its corresponding circumference 2πr (where r ) R sin θ)
7664
Langmuir, Vol. 20, No. 18, 2004
Tadmor
results in an area increment of 2πrR dθ ) 2πR 2 sin θ dθ. Integrating over θ results in
AC ) 2π
θ R 2 sin R dR ) 2πR 2(1 - cos θ) ∫R)0
(A1.7)
Note that for a fixed volume, the radius r is solely a function of θ.
We write for the area dAP d(πr2) dr ) ) 2πr dθ dθ dθ
Putting eq A1.5 into eq A1.7 we get
[
]
π AC ) 2πV 2/3 (2 - 3 cos θ + cos3 θ) 3
-2/3
(1 - cos θ) (A1.8)
Now that we have both AC and AP expressed in terms of constant volume as a function of only the angle of contact, θ, we can use the chain rule to find dAC/dAP
( )/( )
dAC dAC ) dAP dθ
dAP dθ
(A1.9)
After performing both derivatives, dividing dAC/dθ by dAP/dθ and rearranging we get
We write for the circumference dr dL d(2πr) ) ) 2π dθ dθ dθ
( )/( ) ( )/(
dAP dr dr 1 ) 2π 2πr ) dθ dθ dθ r
dL dL ) dAP dθ
Appendix 2: Calculating dL/dAP with the Constrain of Constant Volume
( )/( ) dAP dθ
)
(A2.4)
The geometrical considerations described in Appendix 1 show that at constant drop volume, V, we get (cf. eq A1.5 in Appendix 1)
R)
dL dL dθ dL ) ) dAP dθ dAP dθ
(A2.3)
Putting eqs A2.2 and A2.3 in eq A2.1 we get
dAC/dAP ) cos θ
dL/dAP ) ? Consider a sphere of radius R that is cut by a plane that forms a circle of radius r in that sphere (see Figure 5). The derivative that we want to find is the derivative of a circle circumference (2πr) by circle area (πr 2), and we need to do this derivative under the constrain of constant cap volume (so the angle is free to vary creating different cap shapes, but all have the same volume). To do this we derivate both L and AP by θ and then divide one by the other using the chain rule
(A2.2)
(
)
V 3 π 2 - 3 cos θ + cos3 θ
1/3
(A2.5)
Using the relation r ) R sin θ, we obtain
r)
-1/3
[3Vπ (2 - 3 cos θ + cos θ)] 3
sin θ (A2.6)
Substituting eq A2.6 into eq A2.4 we finally get
(
)
π 2 - 3 cos θ + cos3 θ dL ) dAP 3 V sin3 θ
(A2.1) LA049410H
1/3
(A2.7)
