We analyse the anisotropic thermal expansion properties of a two-dimensional structurally rigid construct made from rods of different materials connected together through hinges to form triangular units. In particular, we show that this system may be made to exhibit negative thermal expansion coefficients along certain directions or thermal expansion coefficients that are even more positive than any of the component materials. The end product is a multifunctional system with tunable thermal properties that can be tailor-made for particular practical applications.
The extent to which materials and structures deform when subjected to changes in temperature is a subject that has been studied for many years. Scientists and engineers must constantly account for temperature effects in their designs, as their neglect could result in various problems (Taylor 1998). Considerable advances are being made in the design, study and manufacture of materials and structures having very particular coefficients of thermal expansion, including materials and structures exhibiting negative coefficients of thermal expansion, i.e. materials which contract when heated (Lakes 1996; Mary et al. 1996; Sigmund & Torquato 1996, 1997; Taylor 1998; Evans 1999; Milton 2002; Sleight 2002; Vandeperre et al. 2002; Vandeperre & Clegg 2003; Barrera et al. 2005; Smith et al. 2005) and in recent years, various composites having predetermined coefficients of thermal expansion have been developed and are already in use in small-scale or high-tech applications that require a good match of the thermal properties (e.g. in teeth fillings (Versluis et al. 1996) or electronic applications (Holzer & Dunand 1997)). Nevertheless, there is still the need to develop simpler and cheaper methods for achieving the same effect on any scale, particularly on a large scale.
Here, we discuss the properties of a simple structure (figure 1) which can be constructed at any length scale and exhibits the very interesting property that its thermal expansion coefficient can be controlled and adjusted to any pre-desired value. As illustrated in figure 1, in its most general case, this structure can be described as a two-dimensional periodic network made from three sets of rods of different materials (materials 1–3).
2. Model for the thermal expansion properties at a temperature T0
We will assume that rods of the same materials are aligned parallel and equidistant from each other in such a way that the three sets of rods intersect through a ‘pin joint’ with each other to form triangles with side lengths l1, l2 and l3 (where lm corresponds to the length of the side made from material m) as shown in figure 1. It has recently been shown that triangles having one side made from a different material than the other two sides can be made to exhibit negative thermal expansion (Vandeperre et al. 2002; Vandeperre & Clegg 2003; Smith et al. 2005) and as we will show, the presence of these triangular units in our construct, potentially having all three sides with different lengths lm and made from different materials, is the key requirement for enabling full control of the thermal expansion properties.
The geometry of our construct may be described in terms of a parallelogrammic unit cell that contains two triangles. If the structure is aligned in space in such a way that the rods of material 2 are always parallel to the Ox2 direction, then the unit cell will have unit cell vectors a=(X11, X12) and b=(0, X22), where X11, X22 and X12 are given by(2.1)(2.2)(2.3)
Taking the most general case in which the materials have different coefficients of thermal expansion αSm, when the structure is subjected to a change in temperature dT, the lengths lm will vary by different amounts defined by dlm=lmαSmdT. These changes in lm owing to a temperature variation will result in changes in size and shape of the macrostructure which may be quantified through the symmetric tensor αij (i, j=1, 2) that describes the anisotropic thermal expansion of a two-dimensional system (Nye 1957). The values of αij are defined as(2.4)where ϵ11 and ϵ22 are the axial strains in the Ox1 and Ox2 directions, respectively, while ϵ12 and ϵ21 are equal to half the shear strain γ. These strains are given by(2.5)
Thus, since(2.6)from equations (2.1)–(2.6), the elements of the thermal expansion tensor αij can be simplified to(2.7)(2.8)and(2.9)
In addition, using standard axis transformation techniques (Nye 1957), we may also obtain α(ζ), the coefficient of thermal expansion in a direction subtending an angle ζ to the Ox1,(2.10)
Furthermore, using the standard theory of principal strains (Gere 2001), we can identify the maximum and minimum thermal expansion coefficients afforded by the structure, which are given by(2.11)that occur at mutually orthogonal directions which are oriented at an angle of ζmax/min to the Oxi axes where ζmax/min is given by(2.12)
3. Results and discussion
Equations (2.7)–(2.10) suggest that, in general, the coefficient of thermal expansion α(ζ) will depend on the following:
the geometry of the system (i.e. the relative magnitudes of lm),
the properties of the materials (i.e. the magnitudes of αSm), and
the direction of measurement (i.e. the angle ζ).
These equations also suggest that the change in shape and size of the macrostructure owing to a change in temperature may result in negative thermal expansion (i.e. thermal contraction) in certain directions. For example, assuming that the macrostructure is constructed using conventional materials having positive but different coefficients of thermal expansion, the structure will contract in the Ox1 direction when heated (i.e. α11 will assume negative values) if . Equations (2.11) and (2.12) suggest the directions where the structure exhibits maximum/minimum thermal expansion coefficients and that the signs/magnitudes of these coefficients depend on the magnitudes of αij (equations (2.7)–(2.9)), which in turn depend on the relative lengths and coefficients of thermal expansion of the rods. All these suggest that the thermal expansion properties may be completely controlled through the choice of the parameters lm and αSm, and as a consequence, by careful choice of these parameters, one may engineer macrostructures that exhibit predetermined thermal expansion properties, thus enabling the construction of systems which are tailor-made for particular practical applications.
These equations also show that, in general, α12 is not zero and hence the macrostructure may shear when subjected to a change in temperature. The condition for the structure not to shear is , and this can be satisfied by the trivial solution where all αSm have the same value, αS. In fact, under such conditions, the macrostructure will be isotropic in-plane, with α(ζ)=αS, and upon heating the macrostructure will expand while maintaining its aspect ratio in accordance with common expectation. Another system that will not shear is the special case when αS1=αS3 and the triangles are isosceles with l1=l3 as discussed further on.
In an attempt to understand more clearly the potential of this macrostructure as a system with variable thermal expansion and, in particular, as a system that can exhibit thermal contraction, we shall consider the following five special cases.
Equilateral triangles with αS1=αS3≠αS2.
Equilateral triangles with αS1=αS2≠αS3.
Equilateral triangles with αS1≠αS2≠αS3≠αS1.
Isosceles triangles with l1=l3≠l2 and αS1=αS3≠αS2.
More general cases.
(a) Case I: special case when the triangles are equilateral and αS1=αS3≠αS2
Let us first consider the particular case where triangles are equilateral (having lm=l) at the temperature of interest and the intrinsic coefficients of thermal expansion are such that αS1=αS3≠αS2 (i.e. the rods aligned parallel to the Ox2 direction have a thermal expansion coefficient that is different from the other two sets of rods).
In such a case, as illustrated in figure 2, the thermal expansion coefficients of the macrostructure will depend on the direction of measurement (i.e. it is anisotropic) and the particular values of αS1 and αS2. (As stated earlier, in the special case when αS1=αS2=αS3, the system will become isotropic where α(ζ)=αS1=αS2=αS3 as expected from a system made from a single isotropic material. In this respect, we note that isotropic behaviour can only be achieved when all rods are made from the same material.)
More specifically, when lm=l and αS1=αS3≠αS2, equations (2.7)–(2.12) for the on-axis and off-axis values of the coefficient of thermal expansion for this system simplify to(3.1)(3.2)(3.3)(3.4)(3.5)(3.6)
Equation (3.3) suggests that this system will not shear when heated, while equation (3.5) indicates that the maximum/minimum thermal expansion will always occur on-axis. Furthermore, from these equations, we can deduce the following.
When αS1=αS3<αS2, minimum thermal expansion is exhibited in the Ox1 direction. This becomes negative when 4αS1=4αS3<αS2. The magnitude of the negative thermal expansion can be increased by increasing αS2 relative to αS1=αS3 (figure 2a). In addition, when αS1=αS3<αS2, the thermal expansion coefficient in the Ox2 direction α22=αS2 is the maximum (most positive) thermal expansion coefficient exhibited by the system.
When αS1=αS3>αS2, maximum thermal expansion is exhibited in the Ox1 direction where the maximum thermal expansion is greater than any of the individual αSms (figure 2b). In addition, in this case, where αS1=αS3>αS2, the thermal expansion coefficient in the Ox2 direction α22=αS2 is the minimum (least positive) thermal expansion coefficient exhibited by the system.
All this is consistent with the work done by Smith et al. (2005). An animation illustrating this type of behaviour is supplied in the electronic supplementary material.
(b) Case II: special case when the triangles are equilateral and αS1=αS2≠αS3
It is important to note that the system in case I exhibits zero α12 and its maximum/minimum thermal expansion occurs on-axis due to the fact that the system is aligned in such a way that the rods which have a different thermal expansion from the other two sets (i.e. the rods of material 2) are set to be aligned parallel to the Ox2 direction (i.e. the Ox1 and Ox2 directions correspond to lines of symmetry). In fact, when this is not the case, for example, if the thermal expansion coefficients of the rods were such that αS1=αS2≠αS3, then equations (2.7)–(2.12) simplify to(3.7)(3.8)(3.9)(3.10)(3.11)(3.12)
Equation (3.9) suggests that, in this case, the shear component is not zero and the extent of shear deformation when the structure experiences a temperature change is proportional to the differences between the thermal expansion coefficients of rods of materials 1 and 3 (assuming that the rods of material 2 remain aligned with the Ox2 direction). This can be attributed to the fact that the Ox1 direction does not correspond to a line of symmetry as was the case in case I. In addition, from equations (3.7) and (3.8) we may deduce that for this system, the on-axis thermal expansion coefficients will never be negative. However, negative thermal expansion is still possible and, as illustrated from equation (3.11), the directions of maximum/minimum thermal expansion coefficients are at −30° to the Ox1 and Ox2 axes, i.e. in directions that are normal and orthogonal to the rods which have a different thermal expansion from the other two sets (i.e. the rods made from material 3). In fact, if αS3>4αS1=4αS2, then the system will exhibit negative thermal expansion which is at a maximum at −30° to the Ox2 direction, i.e. a direction which is orthogonal to the rod of material 3. Note that these conclusions can also be reached by considering that the system in case II is the equivalent of the system in case I after this is rotated by 60°. In fact, the plots of α(ζ) for this system (not shown) will be equivalent to the ones in figure 2 with the difference that they would be 60° out of phase. A similar discussion can be made for the system containing equilateral triangles where αS2=αS3≠αS1.
(c) Case III: special case when the triangles are equilateral and αS1≠αS2≠αS3≠αS1
If we were to consider a more general scenario where the triangles are still equilateral at the temperature of interest but the three sides have different thermal expansion coefficients (i.e. αS1≠αS2≠αS3≠αS1), then equations (2.7)–(2.12) will simplify to(3.13)(3.14)(3.15)(3.16)(3.17)(3.18)
These equations suggest that as in case II, this structure will shear when it experiences a change in temperature where the extent of shear deformation is proportional to the differences between the thermal expansion coefficients of materials 1 and 3 (assuming that the system is still aligned in such a way that the rods of material 2 are always parallel to the Ox2 direction). This shearing can once again be attributed to the fact that the Ox1 direction does not correspond to a line of symmetry.
In addition, we note that the system is still capable of exhibiting negative thermal expansion, even on-axis. For example, for negative thermal expansion in the Ox1 direction, we require that αS2>2αS1+2αS3. However, in this case, minimum/maximum thermal expansion coefficients will not be exhibited on-axis and in fact the directions of maximum/minimum thermal expansion α(ζ) will occur in the directions which are orthogonal to each other and at an angle of ζmax/min to the Oxi axis, where ζmax/min is given by equation (3.17). Typical plots of α(ζ) for various combinations of the intrinsic thermal expansion coefficients are shown in figure 3. These plots highlight the increased anisotropy (when compared with case I, see figure 2) in the thermal expansion properties that results from using three materials having different thermal expansion coefficients rather than just two.
(d) Case IV: special case when the triangles are isosceles with l1=l3≠l2 and αS1=αS3≠αS2
For a given set of rods with αS1=αS3≠αS2, one can obtain a greater range of thermal expansion coefficients than those in case I by relaxing the condition that all the lengths lm are initially equal. To investigate this systematically, we will now consider the special case when the triangles are isosceles rather than equilateral with l1=l3≠l2 and αS1=αS3≠αS2. In such cases, equations (2.7)–(2.12) will simplify to(3.19)(3.20)(3.21)(3.22)(3.23)(3.24)
From these equations, one may note that although this system shares various properties with the simpler system in case I where the triangles are equilateral (e.g. system does not shear and maximum/minimum thermal expansion occurs on-axis), in the current case, more extreme properties can be observed. In particular,
when αS1=αS3<αS2, minimum thermal expansion is exhibited in the Ox1 direction (as was the case for equilateral triangles), where the minimum value of the thermal expansion may be lowered by (a) increasing the value of αS2 relative to αS1=αS3 (as was the case for the equilateral triangles), or (b) increasing l2 relative to l1=l3. In fact, negative thermal expansion in the Ox1 direction can now be obtained when rather than αS2>4αS1, as was the case for equilateral triangles. This is very significant, particularly from a manufacturing point of view, since as the ratio l1/l2 becomes smaller, it becomes possible for a structure to exhibit negative thermal expansion even if it is made from materials which have different though comparable values of αSm. Furthermore, we note that very large negative thermal expansion coefficients can be exhibited in the limit when l2→(l1+l3) and
when αS1=αS3>αS2, maximum thermal expansion is exhibited in the Ox1 direction (as was the case for equilateral triangles), and this maximum value may be increased by (a) decreasing the value of αS2 relative to αS1=αS3 (as was the case for the equilateral triangles), or (b) decreasing l2 relative to l1=l3. Note that very large positive thermal expansion coefficients can be exhibited in the limit when l2→(l1+l3), thus providing the useful property of thermal strain magnification, i.e. a system which exhibits very high strains for small changes in temperature. This can have very important applications, for example, as components of mechanical systems that respond to very small temperature changes.
Typical plots of α(ζ) for various combinations of the intrinsic thermal expansion coefficients are shown in figure 4. These plots highlight the wider range of possible values that the thermal expansion coefficients of this system can assume when compared with the system in case I (figure 2).
(e) Case V: more general cases
Although the special cases I–IV described above highlight the versatility of this structure, we note that equations (2.7)–(2.12), which apply to more general cases, suggest that, in reality, the manufacturer of such systems can control the properties of these systems through choice of any of the independent variables of the system (the relative lengths and the relative intrinsic thermal expansion coefficients of the materials used). In this respect, it is important to note that since lm can assume a continuous range of values, in practice, it would be easier to fine-tune the thermal expansion properties of the structure by altering the magnitudes of lm rather than by varying αSm, since the range of values that αSm can assume are discreet and limited to those of available materials.
Typical plots of α(ζ) for various combinations of lengths and the intrinsic thermal expansion coefficients are shown in figure 5. These clearly illustrate the versatility, anisotropicity and tunability of this connected rods system. An animation illustrating this more generalized behaviour is also supplied in the electronic supplementary material.
4. The temperature dependence of the thermal expansion coefficients
Referring to equations (2.7)–(2.12), it should be noted that since the values of the thermal expansion coefficients of the structure depend on the lengths lm, in cases when at least one of the materials has a different αSm from the other two, the values of the thermal expansion coefficients of the structure will be dependent on the temperature since the relative magnitudes of lm are themselves dependent on the temperature. In fact, it is important to note that the thermal expansion properties discussed above are only valid for small temperature changes.
This temperature dependence of the coefficients of thermal expansion can have some very interesting consequences which, for example, may be illustrated by considering a special case of case I where at a temperature T0, the triangles are equilateral with αS2=4αS1=4αS3. From equation (3.1), this system will exhibit zero thermal expansion α11 at T=T0. However, as the temperature is increased, the lengths lm will increase in such a way that l2 will always be longer than l1 and l3 for temperatures T>T0, and conversely, if the temperature is decreased, the lengths lm will decrease in such a way that l2 will always be shorter than l1 and l3 for temperatures T<T0. In such cases, when T≠T0, the system will no longer be represented by case I but by case IV, i.e. although the system exhibits zero thermal expansion α11 at T=T0, it will exhibit negative α11 when T>T0, which becomes more negative as T is increased, and positive α11 when T<T0.
In this study, we have modelled a system constructible at any scale, with adjustable thermal expansion. This system can exhibit very interesting and useful properties including thermal contraction (negative thermal expansion) and extreme thermal expansion properties, for example, a positive thermal expansion coefficient which is much more positive than any of the component materials. We have also shown that the thermal expansion properties of this system are highly anisotropic and temperature dependent. Given the simplicity of the construction (compared to other systems which can exhibit similar properties (Lakes 1996; Sigmund & Torquato 1996, 1997; Milton 2002)), its adjustability and structural rigidity (since the construct under analysis consists of triangles which confer substantial structure rigidity), we envisage that the proposed construct or variations of it (including three-dimensional constructs where rods of different materials now form, for example, the edges of tetrahedra) should find extensive use in many practical applications.
We acknowledge the financial support of the EU (through the FP6 CHISMACOMB project) and that of the Malta Council for Science and Technology (through its RTDI programme).