OPTICALMEASUREMENT OF SOAPFILM THICKNESS
200 1
Stokes’ Principle of Reversion and the Optical Measurement of Soap Film Thickness by J. B. Rijnbout V a n ‘t Hog Laboratory, Sterrenbos 19, Utrecht, The Netherlands (Received November 10, 1969)
A general equation is derived for the reflection of light by a soap film as a function of its thickness, without assuming any model for the surface layers. It is also shown how the thickness of the aqueous core of the film may be obtained from experimentaldata. The results agree with Corkill’s infrared absorption measurements.
Introduction The intensity of monochromatic light reflected by a soap film (as found in soap bubbles) is often used to calculate its thickness. The connection between the coefficient for the reflected intensity, R2, and the thickness, d, is given by Airy’s formula’ 4r2 sin2 0 R2 = (1 - Y ~ )4r2 ~ sin2 /3
+
where r2 is the coefficient for the intensity reflected by a single surface and 0 = kd COS 8 with IC the wave number (2anlk) within the film and 8 the angle of refraction in the film. This equation applies to a plane-parallel homogeneous layer. We know, however, that a soap film is in fact not homogeneous but that its surfaces are covered with a layer of soap molecules. To take this into account, Duyvis,2 Lyklema, Scholten, and A/Iyselslaand Frankel and Mysels4considered a film consisting of three homogeneous layers, i.e., two soap layers and an aqueous core. An interesting result of their calculations is that the difference between the thickness so calculated and that calculated from eq 1 is independent of the core thickness. In order to check if this is valid for any model of the surface layer, a more general derivation would be desirable. In searching for such a proof I found that in principle it was given already by Stokes in his 1849 paper6 on the perfect blackness of the central spot in Newton’s rings. I propose to give this derivation in modified and extended form and to apply the resulting equation to some model systems.
General Formulation of Reflection and Transmission at an Interface Consider the interface between two media, called 1 and 3. Let the nonabsorbing transition layer from medium 1 to medium 3 be included between two mathematical planes situated in the homogeneous parts of 1 and 3 (see Figure 1). If a light beam of unit ampli-
tude, polarized either parallel or perpendicular to the plane of incidence, is incident on 0, the reflected amplitude may be written as
r
=
IrI exp(-ip)
(2)
where p is the phase change on reflection and i = and the transmitted amplitude is likewise given by
43,
t = It1 exp(-iT)
(3)
The phase change p refers to 0, the point where the incident beam meets the first boundary plane, and r refers to 0’, the point opposite to 0 on the other boundary plane; this convention about the phase reference points applies throughout this paper. For a beam which is incident from medium 3 and makes the same angle with the normal as the transmitted beam of Figure 1, the reflection and transmission coefficients are called r’and t’. Nom according to Stokes, in the absence of absorption the time may be reversed in Figure 1 without effect on the amplitudes. So in a thought experiment all arrows may be reversed and r and t replaced by their complex conjugates r* and t*, as the amplitude which is leading in phase when time goes forward, becomes the lagging one when time goes backward. (Compare Figure 1with Figure 2A.) But the situation of Figure 2A can also be obtained by superposihion of the separate beams r* and t* and their respective reflected and transmitted beams (Figures 2B and 2C). In mathematical form
r*r
+ t*t‘
=
1
(4)
r*t
+ t*r’
= 0
(5)
(1) See, for instance, M . Born and E. Wolf, “Principles of Optics,” Pergamon Press, London, 1959, p 324. (2) E. M. Duyvis, “The Equilibrium Thickness of Free Liquid Films,” Thesis, Utrecht, 1962. (3) J. Lyklema, P. C. Scholten, and K. J. Mysels, J. Phys. Chem. 69, 116 (1965). (4) 9. P. Frankel and K. J. Mysels, J . AppZ. Phys., 37, 3725 (1966). (5) G . G. Stokes, ‘”lathematical and Physical Papers,” Vol. 11, Cambridge University Press, 1883, p 89.
Volume 74, Number 9 April SO, 1970
2002
J. B. RIJNBOUT 3
21
I2
1
I I
I I I
I
I
I
I
I I I
I I I I I I
I
I
I
AI Figure 1. Reflection and transmission at an interface: 1, incident beam; r reflected beam; t , transmitted beam; 0, phase reference point for medium 1 ; 0’, phase reference point for medium 3.
Figure 3. Reflection and transmission by a film: 1, external medium; 2, surface layer; 3, core.
tensity reflection coefficient is the same in both directions. So we find 1
2
1 ;
3
r
=
1 - rl*r’
-.--r’*t‘
=
t1*
(7)
t’*
These general relations are valid for a layer of arbitrary thickness and refractive index profile and may now be used in the calculation of film reflectivity.
General Formulation of Reflection by a Film We consider a soap film, consisting of an optically homogeneous central layer of thickness & with two identical surface layers (Figure 3). By summation of amplitudese we find for the amplitude reflection coefficient of the film
A
1 2 3
1 2 3 I
1
I
’
R = r f
I I
r’ =
C
Figure 2. A, the beams of Figure 1 after time inversion; B, C, the separate contributions of r* and t* to the beams of Figure 2A.
These equations give
-.-r*t t*
’
t‘
=
1 - r*r t*
(6)
We find r and t in terms of r’ and t’ by starting with a primary beam incident from the other side or by solving eq 4 and 5 for r and t (via r* and t*). In the latter case we notice that from eq 4 t*t’ is real, thus t*t’ = tt’* or t/t* = t‘/t‘* meaning that the phase shift on transmission is the same in both directions, and that by multiplying the first eq 6 by its complex conjugate we obtain the result r‘rl* = rr* meaning that the inThe Journal of Physical Chemistry
(9)
exp(-ip’)
[rl/
we find for the reflected intensity, multiplying R by its complex conjugate
RR* =
(8)
1 - (r’)2exp(-2if13)
where f13 = k3d3 cos O3 with leI = wave number in the central layer, d3 = thickness of the central layer, and B3 = angle of refraction in the central la,yer. Converting in this equation r and t to r’ and t‘ by eq 7, remembering that from eq 6 r’*r’ equals r*r and defining pl by
t’
B
tr’t’ exp(-2ip3)
=
(1-
+
4rr* sinZ(p3 p’) r ~ * ) ~4rr* sin2 (pa
+
+ p‘)
(10)
+
This conforms to eq 1, with p replaced by p3 p‘. Again rr* is the intensity reflection coefficient of a single surface which may be measured separately or obtained from the maximum value of RR* which equals 4rr*/ (1 rr*)2. From the argument of the sine in eq 1 one obtains a film thickness d = p/le cos e. If one uses the bulk refractive index n3 for the calculation of k cos 0 this thickness is the so-called equivalent water thickness
+
(6) See, for inetanoe, ref 1, p 323.
OPTICALMEASUREMENT OF SOAPFILM THICKNESS
2003
d, of the film. So we have for a film with known reflectivity, equating the right-hand members of eq 1 and 10
the film is the same for all angles of incidence and has the form given by Duyvis2and by Frankel and Mysels4 for normal incidence. Extension to Films with Heterogeneous Surface Layers. Equation 15 can be written as
k3d, cos O3 =
p3
+ p’
f
mln
p’ f mln
=
dw
d3 +
(11)
k3 cos O3
where ml is an as yet unknown whole number. Since p’ is independent of ti3, we find indeed a constant difThe magnitude of this ference between d3 and d,. difference, however, can only be calculated by the introduction of a model for the surface layer. Application to Films with Thin Homogeneous Surface Layers. We take first the model of a thin homogeneous surface layer 2 , between an external medium 1 and a central layer 3. The phase shift p’ for reflection from the inside against such a layer is given by7 tan p‘ = rzl(l
raz(1
- r32’)
+ rziz) +
sin (2k2dzcos OZ) rzi(1 r3z2) COS (2kzdz COS Oz)
+
(12)
where rd,is the amplitude reflection coefficient for light going from medium i to medium j . As we are interested in surface layers having a thickness small compared to the wavelength of light, we expand eq 12 up to the first power of lc2dz putting the tangent and the sine equal to their arguments and the cosine equal to one. Furthermore, in order to calculate rij we now have to make a choice with respect to the state of polarization of the incident light. We limit the treatment to the situation more commonly encountered in practical measurements because any other choice disturbs the simplicity of the resulting equations, and substitute the Fresnel expressions for light polarized with its electric vector perpendicular to the plane of incidence rdJ =
ndcos Ba - n j cos 0, n, cos Oa n, cos Oj
+
(13)
where ndis the refractive index of medium i. Then we find pl
=24
nz2- n12 cos ea f mzn ng2- n12
(14)
where m2is a whole number, and substituting this in eq 11 d, =
d3
nz2- n12 + 2dzn32 - nI2
(15)
I n order that d, - d3 goes to zero with dz, ml, and m2 must satisfy the condition ml m2 = 0. If for a homogeneous film (dz = 0, d, = d3) we have p’ = 0, then m2 = 0 (eq 14) and ml = 0 (eq 11). So for a thin homogeneous surface layer and perpendicular polarization of the light, the difference between the core thickness and the equivalent water thickness of
+
(n32- nI2)d, = (n.12 - n12)dt (16) where the summation extends over all layers. The form of this equation suggests that it will apply to any number of layers, which can indeed be proved by evaluating p’ from the characteristic matrix* of a multilayer surface, again assuming that the thickness of this compound surface layer is small with respect to the wavelength of the light used. Franliel and Mysels obtained the same result in another way, restricted to the case of normal incidence. To make further use of eq 16 we need pairs of values for nl and di which are generally unknown and difficult to estimate. In the simplest and practically most important case the external medium is air with nl = 1. In that case we need values of ni2 - 1 which may be obtained from the specific refraction A , of the substance of layer i as defined in the Lorentz-Lorens equation
where p i is the density of the ith layer. (The applicability of this equation to thin layers instead of homogeneous bulk material may be doubted, but i t probably is not far from the truth and a better approximation is not available.) We substitute ni2 - 1 from eq 17 into eq 16 and for p & we write m,, the mass of layer i per cm2 of surface area. Then we obtain 1
d3 = d, - n32 ---~1(nd2 -1
+ 2)A,mi
where the prime indicates that the central layer is excluded from the summation. As a reasonably accurate value of n2 2 can be obtained from a carefully estimated value of n, the second term in eq 18 may now be evaluated when the composition of the surface layer is known, without need for an accurate knowledge of its thickness and refractive index.
+
Comparison with Absorption Masurements To determine the water content of very thin soap films, Corkill, et al.,9 measured their infrared absorption at 2.93 pm, a method which avoids the difficulties due t o the presence of surface layers, but which is subject to uncertainty as to the value of the molar extinction coef(7) See ref 1, p 61. (8)See ref 1, p 54. (9) J. M. Corkill, J. F. Goodman, and C. P. Odgen, Trans. Faraday Soe., 61, 583 (1965). Volume 74, Number 9
April $0,1970
J. B. RIJNBOUT
2004 ficient of water in these thin layers. The reflectances and the amount of soap per cm2of surface area of these films of decyltrimethylammonium decyl sulfate (CloClo) are known so m7e can also apply our method to them and compare the results. From tracer measurements and surface tension os. concentration curves Corkill, et aZ.,1° found that the surface concentration in these filmsocorresponds to one molecule of Cd2lo per (58 f 3) A2, which means a mass mt of (1.25 f 0.06)10-' g/cm2. Now we need an estimate of the refractive index of the surface layer to calculate nt2 2. We may think of this layer as consisting of a paraffinic part containing the hydrocarbon chains and a part containing the ionized groups in an aqueous environment. The refractive index of the hydrocarbon part will be lower than that of bulk decane which equals 1.41 at a density of 0.73. The lowest possible density of the hydrocarbon layer is obtained if the 14.5 8 long decyl chains are extended perpendicularly to the surface. This would give a density of 0.56 and a refractive index of 1.31. The aqueous part containing the ionized groups will have a refractive index larger than 1.33 but presumably not larger than that of a 40% ammonium sulfate solution (containing about four water molecules per ion) which equals 1.39. So it does not seem necessary to differentiate between the paraffinic layer and the layer of head groups and it is reasonable to assume that the refractive index of the surface layer is between 1.31 and 1.41 and n2 2 = 3.85 =k 0.14. Here we consider the head groups as components of the surface layer whereas the water which is present in the layer of head groups is excluded from the surface. This is possible because the components in a layer contribute additively to the specific refraction of the mixture, and so may be thought of as contributing additively to n2 - 1 (eq 17), whereas n2 2 is determined by the refractive index of the mixture as a whole. If we want to exclude a component from the surface layer, we should exclude its contribution to n2 - 1, but not that to n 2 2 . This is in accordance with the meaning of these factors: n2 - 1 is built up additively from the polarizabilities of the molecules and (n2 2)/3 is the Lorentz correction factor for the internal field, which depends on the average polarizability of the surroundings of a molecule. Further, we need the specific refraction of CloClo. As we do not have data on the density and refractive index of C&lo solutions we have to derive its specific refraction from data on related compounds. The molar refraction of C&lo which equals its specific refraction times its molecular weight is obtained by adding the molar refraction of ClzH250S03Na(sodium dodecyl sulfahe) to that of C12Hzs(CH3)3NBr (dodecyltrimethylaminonium bromide) and subtracting four times the molar refraction of a CH2 group and that of aqueous NaBr. To calculate the specific refraction of a solute we
+
+
+
+
+
The Journal of Physical Chemistry
write the Lorentz-Lorenz equation for a two-component system (n2 - l)/(n2
+ 2)
= &go
+ Alg1
(19)
where A. and A1 are the specific refractions of solvent and solute and go and gl their masses per unit volume of solution. By differentiation we obtain
A1 =
6n dn (n2 212 dg,
+
+ A0 (1 - $)
(20)
+
when the solution density p equals go gl. For micellar solutions of sodium dodecyl sulfate, dn/dgl = 0.1257 crn3/gl1 and dp/dgl = 0.111.12 For dodecyltrimethylammonium bromide, dn/dgl = 0.1542 cm3/g" and dp/dgl = 0.046.13 This gives for sodium dodecyl sulfate A = 0.2539 cm3/g and a molar refraction of 69.2 cm3 and for dodecyltrimethylammonium bromide A = 0.2833 cm3/g and a molar refraction of 87.4 cm3. The molar refraction of a CH2 group is 5.17 cm3 and for aqueous NaBr it is given as 12.8 in Landolt-BOrn~tein.'~So the molar refraction of Cl0Clo becomes 69.2 87.4 - 20.7 - 12.8 = 123.1 cm8 and A = 0.2812 cm3/g. If we put these values of A , m , and n2 2 into eq 18 and take for n3 the refractive index of a 0.5 M KaBr solution as used by Corkill, which equals 1.342, we get
+
+
d3
=
d,
- (33.8 f 3.5) 8
(21)
We obtain d, from the reflectance of the film. For the films used in their absorption measurements, Corkill, et ~ l . give , ~ the ratio of the reflection intensity of the film to that, of the last bright band. From eq 1, substituting @ = ksd, for normal incidence and approximating sin /3 = P for these thin films we obtain
In the calculation of k3 we use n3 = 1.342 and ho = 5461 A, r2 is calculated from the Fresnel expression (eq 13) with n = 1.342 as it is hardly modified by the presence of a very thin surface layer; this gives r2 = 0.021. The values of cl, and da thus obttined are given in Table I. The maximum error of 5 A in d3 is composed of a variable error of at most 1.5 8 in d, and a constant error of at most 3.5 A in the second term of eq 21. (10) J. M. Corkill, J. F. Goodman, C. P. Odgen, and J. R. Tate, Proc. Roy. Soc., A273, 84 (1963). (11) H.V. Tartar and A. L. M. Lelong, J. Phys. Chem., 59, 1185 (1955). (12) K.A. Wright and H. V. Tartar, J. Amer. Chem. Soc., 61, 544 (1939). (13) A. B. Scott and H. V. Tartar, ibid., 65, 692 (1943). (14) Landolt-Bornstein, "Physikalisch-Chemische Tabellen," 5.Aufl. E g IIIb, Julius Springer, Berlin, 1935, p 1701.
OPTICALMEASUREMENT OF SOAPFILMTHICKNESS
2005
These values for d3 may be compared with those given in the last column, which were derived from infrared absorption at 2.93 pm as reported by Corkill, et aL9 As a 0.5 M NaBr solution contains about 50 g of NaBr and 990 g of HzO/l., their values of the thickness were multiplied by 1.01 to obtain the equivalent thickness of a 0.5 n/l NaBr solution. All values of d3 from eq 21 are seen to be higher than the corresponding values
Then these quantities will be related by an equation of the form: A X d 4 r ) = da(eq 21) B. Assuming that d3(ir) and d8 (eq 21) are of about equal precision we find by the usual methods of least squares
Table I: Comparison of Core Thicknesses Obtained from Reflectivity and from Infrared Absorption
104R"&X
Number
eq 22,
da, ir,
A
da, eq 21,
obsd
85 95 105 115 125 135 145 155 165 175 f 5
1 6 11 4 6 2 2 3 1 2
57.3 60.5 63.6 66.6 69.4 72.2 74.8 77.3 79.7 82.2 2~1.5
23.5 26.7 29.8 32.8 35.6 38.4 41 . O 43.5 45.9 48.4 h5.0
22.2 24.3 27.1 29.9 32.7 34.3 35.6 39.7 44.4 48.0
R=
dw,
A
A
of d3 from infrared absorption. Systematic differences between the two may be caused on one side by an incorrect second term in eq 21 which leads to a constant absolute error in d3 (eq 21) and on the other side by an incorrect molar extinction coefficient of water which leads to a constant relative error in da (ir).
+
(1.01
f
O.O3)da(ir) = d3(eq 21)
- (2.6 * 0.9) 8
(23)
The standard deviation from this straight line amounts to 1.0 A for one observation. Due to the fact that the values reported for RZ/R2,,, are grouped dn classes with a half-width corresponding to about 1.5 A we expect a standard deviation of 1.5/2/3 = 0.87 A already from this source only, so there is no reason to look for a better fit. That A is not greatly different from unity indicates that the molar extinction coefficient at 2.93 pm of mater in these films is close to 133, the value used by Corkill, et aL9 The agreement between the two sets of core thicknesses may be improved by changing the second term at the right-hand side of eq 21 to -36.4 A which is still within the range of expectations, and suggests that the surface concentration and refractive index could be a few per cent higher than we assumed. As the sets of values in the last two columns of the table represent completely independent estimates of the water core thickness, their agreement may be considered very satisfactory.
Acknowledgment.
Thanks are due to Professor
J. Th. G. Overbeek and Professor A. Vrij, whose interest stimulated this publication and who contributed to its final form by many helpful suggestions.
Volume 74, Number 9 April 30,1970
