904
Langmuir 1998, 14, 904-909
Elastic Behavior of Anisotropic Monolayers Pirooz Mohazzabi Department of Physics, University of WisconsinsParkside, Kenosha, Wisconsin 53141
Feredoon Behroozi* Department of Physics, University of Northern Iowa, Cedar Falls, Iowa 50614 Received April 9, 1997. In Final Form: October 29, 1997 The theory of elasticity in two dimensions is generalized to include anisotropy. It is found that six coefficients are required to fully characterize the elastic response of anisotropic films. The theory is applied to monolayers with specific boundary conditions which represent cases of particular interest in the study of Langmuir-Blodgett films. In the special case when an anisotropic film possesses two orthogonal symmetry planes, only four constants are needed to fully characterize the elastic behavior of the film. Several experimental procedures are outlined to obtain the elastic constants experimentally.
Introduction
Theory
In a recent paper, we reviewed the theory of elasticity in two-dimensional isotropic systems and applied the results to several specific cases of interest. In particular, the theory was used to explore the elastic behavior of solid Langmuir-Blodgett films under several commonly encountered boundary conditions. In each case the appropriate elastic moduli were derived in terms of the two Lame´ coefficients of an isotropic solid film.1 However, it is well-known that films of long-chain fatty acids exhibit polymorphism, i.e., they can exist in a variety of crystallographic states, some of which show marked directional anisotropy.2-5 Indeed, Brewster angle microscopy, electron diffraction, X-ray diffraction, and atomic force microscopy studies of Langmuir-Blodgett films have revealed that solid films may exhibit a variety of structural anisotropies due to unusual head-group topology, tilt angle of the alkyl chains, and head-group interactions.6-10 As expected, anisotropic solid films cannot be described by only two Lame´ coefficients, and a larger number of independent elastic constants is required for their description due to the lower symmetry. In this work, we generalize the two-dimensional theory of elasticity to include the case of anisotropic films. Again, we apply the results to various special cases and commonly encountered boundary conditions, discuss the relationship among different elastic constants, and describe some of the experimental techniques that may be employed for the measurement of the relevant elastic constants of anisotropic monolayers.
The general form of the free energy of a deformed anisotropic elastic body is given by11
(1) Behroozi, F. Langmuir 1996, 12, 2289. (2) Petty, M. C. Langmuir-Blodgett Films; Cambridge University Press: Cambridge, 1996. (3) Tredgold, R.H. Order in Thin Organic Films; Cambridge University Press: Cambridge, 1994. (4) Knobler, C.M.; Desai, R.C. Annu. Rev. Phys. Chem. 1992, 43, 207. (5) Kaganer, V. M.; Peterson, I. R.; Shih, M. C.; Dubrin, M.; Dutta, P. J. Chem. Phys. 1995, 102, 9412. (6) Overbeck, G. A.; Honig, D.; Wolthaus, L.; Gnade, M.; Mobius, D. Thin Solid Films 1994, 242, 26. (7) Schroter, J. A.; Plehnert, R.; Tschierske, C.; Katholy, S.; Janietz, D.; Penacorada, F.; Brehmer, L. Langmuir 1997, 13, 796. (8) Fryer, J. R.; Hann, R.A.; Eyres, B. L. Nature 1985, 313, 382. (9) Kenn, R. M.; Bo¨hm, C.; Bibo, A. M.; Peterson, I. R.; Mo¨hwald, H.; Als-Nielsen, J.; Kjaer, K. J. Phys. Chem. 1991, 95, 2092. (10) Chi, L. F.; Eng, L. M.; Graf, K.; Fuchs, H. Langmuir 1992, 8, 2255.
F)
1 C 2 ijkl ij kl
(1)
where Cijkl are the components of the rank-four tensor of elastic constants and ij are the components of the (infinitesimal) rank-two strain tensor. Due to the symmetry properties of the strain tensor, the tensor of the elastic constants must obey the following symmetry properties:
Cijkl ) Cjikl ) Cijlk ) Cklij
(2)
The stress components can be obtained from the free energy according to
σij ) ∂F/∂ij
(3)
Therefore, we find
σij )
∂mn ∂ 1 1 C ) Cmnpq + ∂ij 2 mnpq mn pq 2 ∂ij pq
(
)
∂pq 1 C 2 mnpq mn ∂ij )
1 C δ δ + 2 mnpq im jn pq 1 C δ δ (4) 2 mnpq mn ip jq
)
1 1 C + C 2 ijpq pq 2 mnij mn
in which δij is the Kronecker delta, defined by δij ) 1 for i ) j and δij ) 0 for i * j. Upon using the symmetry relation (11) Landau, L. D.; Lifshitz, E. M. Theory of Elasticity; Pergamon Press: London, 1964; p 37.
S0743-7463(97)00365-X CCC: $15.00 © 1998 American Chemical Society Published on Web 01/16/1998
Elastic Behavior of Anisotropic Monolayers
Langmuir, Vol. 14, No. 4, 1998 905
Cmnij ) Cijmn, eq 4 reduces to the generalized Hooke’s law:
σij ) Cijklkl
(5)
In these equations, the range of each index is three or two, depending on whether the body under consideration is three or two dimensional, respectively. Furthermore, the summation convention is operative, i.e., the need to sum over repeated indices is implied. For an isotropic solid, the tensor of the elastic constants is isotropic, having invariant components in all coordinate systems. Any isotropic rank-four tensor has the following decomposition12
Cijkl ) λδijδkl + µ(δikδjl + δilδjk) + κ(δikδjl - δilδjk)
(6)
for some scalars λ, µ, and κ. However, in the case of the elastic constants for an isotropic medium, the scalar κ must be zero since Cijkl is invariant under the exchange of i and j, but the terms multiplying κ are not. Substituting in eq 5, we obtain
Figure 1. Schematic representation of a two-dimensional film under hydrostatic compression.
even be orthogonal. For notational convenience and following convention, let us recast the matrix of the elastic constants into the following form
[ ]
R λ γ CKM ) λ β δ γ δ µ
σij ) [λδijδkl + µ(δikδjl + δilδjk)]kl ) λδijkk + 2µij (7) which is the generalized Hooke’s law for an isotropic elastic solid, and the scalars λ and µ are the Lame´ constants.13 We now turn our attention to the general anisotropic case in two dimensions. Being a rank four tensor in two dimensions, Cijkl has 16 components. However, due to the symmetry of the stress and strain tensors, namely, σij ) σji and kl ) lk, the number of independent components reduces to 9. Therefore, following the standard threedimensional case, we replace the double-indexed system of stress and strain components by a single-indexed system with a range of three as follows:
σ1 ≡ σ11
1 ≡ 11
σ2 ≡ σ22
2 ≡ 22
σ3 ≡ σ12
3 ≡ 212
σ22 ) λ11 + β22 + 2δ12
(9)
]
(10)
Now let us consider a homogeneous but anisotropic twodimensional elastic film in the x1x2-plane. We first consider the general case with no further reduction in the number of independent elastic constants. Note that in this case, the coordinate axes x1 and x2 bear no relationship to the crystallographic axes of the film, which may not (12) Segel, L. A. Mathematics Applied to Continuum Mechanics; Dover: New York, 1987; p 46. (13) Segel, L. A., Mathematics Applied to Continuum Mechanics; Dover: New York, 1987; p 162.
(12)
σ12 ) γ11 + δ22 + 2µ12 or conversely,
22 )
Furthermore, due to the symmetry Cijkl ) Cklij, we obtain CKM ) CMK, and the number of independent elastic constants reduces to 6. We therefore end up with a 3 × 3 symmetric matrix of the elastic constants:
[
σ11 ) R11 + λ22 + 2γ12
(8)
σK ) CKMM (K,M ) 1, 2, 3)
C11 C12 C13 CKM ) C21 C22 C23 C31 C32 C33
Then the stress-strain relations become
11 )
In other words, the double-index system transforms into the single-index system according to the rule 11 f 1, 22 f 2, 12 f 3 (also note the factor of 2 relating 3 and 33). Then, for example, C1112 becomes C13. With this notation, Hooke’s law is written as
(11)
1 [(βµ - δ2)σ11 - (λµ - γδ)σ22 + (δλ - βγ)σ12] D
1 [-(λµ - γδ)σ11 + (Rµ - γ2)σ22 - (Rδ - γλ)σ12] D (13)
12 )
1 [(δλ - βγ)σ11 - (Rδ - γλ)σ22 + (Rβ - λ2)σ12] D
where D is defined as
D ) (βµ - δ2)R - (λµ - γδ)λ + (δλ - βγ)γ (14) Specific Applications (a) Hydrostatic Compression. For a two-dimensional elastic film under hydrostatic compression, the solid film is compressed uniformly from all sides and no shear stress is involved (see Figure 1). The stress components are therefore given by σij ) - δij p, and eqs 13 for the strain components reduce to
11 ) -
1 [(βµ - δ2) - (λµ - γδ)]p D
22 ) -
1 [-(λµ - γδ) + (Rµ - γ2)]p D
12 ) -
1 [(δλ - βγ) - (Rδ - γλ)]p 2D
(15)
where p should now be interpreted as the two-dimensional analog of the pressure, i.e., force per unit length. In this case, the isothermal compressibility which is generally
906 Langmuir, Vol. 14, No. 4, 1998
Mohazzabi and Behroozi
Figure 2. Schematic representation of a two-dimensional film under uniaxial compression. Note that the sides are free.
Figure 3. Schematic representation of a two-dimensional film under unilateral compression. Note that the sides are clamped.
defined by
In this case, the Poisson ratio, σ(x1), and Young’s modulus, E(x1), are defined, respectively, by
κT ≡ -
1 ∂V V ∂p
( )
(16)
T
reduces to
22 σ11 p and E(x1) ≡ )11 11 11
(23)
λµ - γδ D and E(x1) ) 2 βµ - δ βµ - δ2
(24)
σ(x1) ≡ which reduce to
κT ≡ -
1 ∂A A ∂p
( )
)T
dA/A p
(17)
where A is the area and dA/A is the dilation of the film. In three dimensions, the dialation of an elastic body is, in general, given by14
dV/V ) Tr(ij) ) ii
(18)
which, in two dimensions, reduces to
dA/A ) 11 + 22
(19)
Using eqs 15 for 11 and 22, we obtain
1 dA ) - [µ(R + β) - (δ2 + γ2) - 2(λµ - γδ)]p A D
(20)
which in light of eq 17 results in
(b) Uniaxial Compression (Sides Free). Let the compression take place along the x1 axis and the free sides be in the x2 direction, as shown in Figure 2. Then σ11 ) - p, σ22 ) σ12 ) 0. Consequently, eqs 13 for the strain components reduce to
1 11 ) - (βµ - δ2)p D
12 ) -
1 (λµ - γδ)p D
Similarly, if the uniaxial compression is applied along the x2 axis (with the free sides along the x1 direction), we will have
σ(x2) )
λµ - γδ D and E(x2) ) 2 Rµ - γ Rµ - γ2
11 )
12 )
1 [-(βµ - δ2)p - (λµ - γδ)σ22] D
1 (δλ - βγ)p 2D
(14) Segel, L. A., Mathematics Applied to Continuum Mechanics; Dover: New York, 1987; p 152.
1 [(λµ - γδ)p + (Rµ - γ2)σ22] D
(26)
1 [-(δλ - βγ)p - (Rδ - γλ)σ22] 2D
Solving the second equation for σ22 and substituting in the other two equations, gives
11 ) 12 ) -
(22)
(25)
(c) Unilateral Compression (Sides Clamped). Let the unilateral compression be applied in the x1 direction and the sides in the x2 direction be clamped, as shown in Figure 3. Then σ11 ) - p, σ12 ) 0, and σ22 is determined through the elastostatic equations (eqs 13) by setting 22 ) 0, namely
22 ) 0 )
1 κT ) [µ(R + β) - (δ2 + γ2) - 2(λµ - γδ)] (21) D
22 )
σ(x1) )
[
[
]
2
(λµ - γδ) 1 βµ - δ2 p D Rµ - γ2
]
(λµ - γδ)(Rδ - γλ) 1 δλ - βγ p (27) 2D Rµ - γ2
The unilateral compressional modulus is defined by K′ ≡ σ11/11 ) -p/11. For the unilateral compression applied in the x1 direction, therefore, we obtain
[
K′(x1) ) D βµ - δ2 -
]
(λµ - γδ)2 Rµ - γ2
-1
(28)
Similarly, if the unilateral compression is applied in the
Elastic Behavior of Anisotropic Monolayers
Langmuir, Vol. 14, No. 4, 1998 907
are the reflected ones. The transformation matrix from the unprimed to the primed system is given by
aij ) cos(xi′, xj) )
[
-1 0 0 1
]
(32)
Therefore, using the transformation law for the stress and strain tensors
σij′ ) aimajnσmn and ij′ ) aimajnmn
(33)
σ1′ ) σ1 σ2′ ) σ2 σ3′ ) -σ3 1′ ) 1 2′ ) 2 3′ ) -3
(34)
we find
Figure 4. Reflection of the coordinate system x1x2 through the symmetry plane which is perpendicular to the film surface and contains the x2 axis. The reflected axes are x′1 and x′2.
σ1 ) C111 + C122 - C133
x2 direction, we obtain
[
K′(x2) ) D Rµ - γ2 -
]
(λµ - γδ)2 βµ - δ
2
-1
(29)
The six elastic constants can therefore be determined from the measurements of the isothermal compressibility under hydrostatic compression, Poisson ratios and Young’s moduli for uniaxial compressions in the x1 and x2 directions, unilateral compressional moduli for unilateral compressions in the x1 and x2 directions, and finally some shear measurements, such as the shear modulus σ12/12. However, these measurable elastic constants are not all independent. For example, it can easily be seen from eqs 24 and 25 that
σ(x1) σ(x2)
)
E(x1) E(x2)
)
K′(x1) K′(x2)
)
Rµ - γ2 βµ - δ2
(30)
or by combining the equations for K′(x1), E(x1), σ(x1), and σ(x2), we can show that
E(x1) K′(x1)
Using these results and eq 9 for the primed system with C′KM ) CKM, we find
) 1 - σ(x1)σ(x2)
(31)
Therefore, in choosing the elastic quantities to be measured for the determination of the six elastic constants, care must be taken to ensure their independence.
(35)
On the other hand eq 9 for the unprimed system gives
σ1 ) C111 + C122 + C133
(36)
Comparing these two equations and keeping in mind that the strain components are considered as independent variables here, we conclude that C13 ) 0. A similar argument for σ2 shows that C23 ) 0 also. No further reduction in the elastic constants can be achieved by any other symmetry consideration. Therefore, the matrix of elastic constants reduces to
[
C11 C12 0 CKM ) C21 C22 0 C33 0 0
]
(37)
with four independent elements. It should be noted that these elastic constants further reduce to the two Lame´ constants in an isotropic film as they should. To see this, one can rotate the coordinate system x1x2x3 about the x3 axis through angles of π/2 and π/4 and apply analyses similar to that described above for transformation of the stress components. These, respectively, give the following relationships between the three elastic constants:
C11 ) C22 and C33 )
C11 - C12 2
(38)
Therefore, if we denote C12 by λ and C33 by µ, we obtain Special Case: Films with Two Orthogonal Symmetry Planes We now examine the special, but frequently encountered, case where the film possesses two orthogonal symmetry planes that are perpendicular to the filmsone containing the x1 axis, and the other containing the x2 axisswith different elastic properties in the x1 and x2 directions. This case can be realized, for example, when the crystallographic axes of the film are orthogonal and we choose the coordinate axes along these directions. Of course, the basis of the lattice must also conform to this symmetry. Clearly, reflection of the axes through these symmetry planes leaves the elastic constants invariant. Let us consider the reflection of the coordinate axes through the symmetry plane that is perpendicular to the film and contains the x2 axis (Figure 4). The primed axes
[
λ + 2µ λ 0 λ + 2µ 0 CKM ) λ 0 0 µ
]
(39)
which generates exactly the same stress-strain relationships as those given by eq 7 for a two-dimensional isotropic elastic solid.1 In terms of the notations employed in eq 11 for the elastic constants, we see that for the two-dimensional anisotropic film with two orthogonal symmetry planes, we have γ ) δ ) 0, and the matrix of the elastic constants reduces to
[ ]
R λ 0 CKM ) λ β 0 0 0 µ
(40)
908 Langmuir, Vol. 14, No. 4, 1998
Mohazzabi and Behroozi
and the stress-strain relations (eqs 12 and 13 ) become
σ11 ) R11 + λ22 σ22 ) λ11 + β22
11 and 22 are both negative, therefore, R > λ and β > λ. Finally, for an isotropic film we have R ) β ) λ + 2µ, and all quantities reduce to their correct corresponding values in all cases.
(41)
σ12 ) 2µ12
Experimental Determination of the Elastic Constants
and
βσ11 - λσ22 11 ) Rβ - λ2 22 )
-λσ11 + Rσ22 Rβ - λ2 12 )
(42)
σ12 2µ
Similarly, all of the expressions for the measurable elastic constants that were discussed earlier reduce to the following simple forms:
κT )
R + β - 2λ Rβ - λ2
(43)
(hydrostatic compression) λ σ(x1) ) , β
E(x1) )
Rβ - λ2 β
(44)
In general, six independent measurements are needed to determine the six elastic constants of an anisotropic film. As mentioned earlier, care must be taken to ensure that the measured elastic moduli are independent of one another. More specifically, eqs 30 and 31 give three relations among six of these measurable quantities. Consequently, only three of the six need be measured. To complete the characterization of the elastic response, one must also measure the isothermal compressibility, κT, and two independent sheer moduli of the film. In the special case when the film possesses two orthogonal planes of symmetry, the number of independent elastic constants reduces to four. Thus only four independent measurements are needed to characterize the elastic behavior of the film. In this case the experimental determination of the elastic constants proceeds in a fashion similar to that described for the isotropic film1 with a few modifications. The π-A isotherm and its slope, dπ/dA, can be obtained for the state of unilateral compression in the x1 direction, using a typical Langmuir trough. Here, π is the two-dimensional analog of the pressure and A is the area per molecule. We have
(uniaxial compression in x1 direction) λ Rβ - λ2 σ(x2) ) , E(x2) ) R R
(45)
K′(x1) dπ dπ dπ p ) ) )) dA dA 11A 11A A A A
(49)
(uniaxial compression in x2 direction) K′(x1) ) R (46) (unilateral compression in x1 direction) K′(x2) ) β
For normal materials that shrink axially and expand laterally under the influence of a uniaxial compressional stress and vice versa (and shrink in all directions under hydrostatic compression), the elastic constants of the anisotropic film with two orthogonal symmetry planes, R, β, and λ, must satisfy certain relationships. These are R > λ, β > λ, and λ > 0, which can be established as follows. For these materials, isothermal compressibility, Poisson’s ratios, Young’s moduli, and unilateral compressional moduli are all positive. Consequently, according to eqs 46 and 47, R > 0 and β > 0, respectively, and hence from eqs 44 or 45, λ > 0 and Rβ - λ2 > 0. For the state of hydrostatic compression, eqs 42 reduce to
β-λ p Rβ - λ2
22 ) -
R-λ p Rβ - λ2
K′(x1) ) A
(47)
(unilateral compression in x2 direction)
11 ) -
where K′(x1) is the unilateral compressional modulus in the x1 direction. Therefore, K′(x1) can be obtained from
(48)
Since p > 0 and Rβ - λ2 > 0, and since in these equations
dπ (dA )
(50)
exp
where the subscript “exp” stands for “experimental”. Similarly, measurement of the π-A isotherm for the state of unilateral compression in the x2 direction results in the determination of K′(x2). Since R ) K′(x1) and β ) K′(x2), R and β are experimentally determined. Using the fairly recently developed Langmuir trough with a circular geometry,15 the π-A isotherm and its slope can be determined under conditions of hydrostatic compression. Then the isothermal compressibility can be obtained from eq 17,
κT ) -
dA 1 1 dπ dA 1 ) ) p A dπ A A dA
( )
( )
-1
exp
(51)
With the values of R and β determined earlier, eq 43 yields the value of λ. Therefore, R, β, and λ can all be determined experimentally from the π-A isotherms obtained in Langmuir trough experiments. Determination of µ, however, re(15) Bohanon, T. M.; et al. Rev. Sci. Instrum. 1992, 63, 1822.
Elastic Behavior of Anisotropic Monolayers
quires shear experiments.16 This is in sharp contrast to the case in isotropic systems where the two elastic constants, λ and µ, can be determined from compressional (16) Abraham, B. M.; Ketterson, J. B.; Behroozi, F. Langmuir 1986, 2, 602.
Langmuir, Vol. 14, No. 4, 1998 909
data in a Langmuir trough under two different boundary conditions. Acknowledgment. F.B. wishes to thank the Carver Trust for supporting this work. LA9703656
