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Ind. Eng. Chem. Res. 2000, 39, 654-665
An Auxetic Filter: A Tuneable Filter Displaying Enhanced Size Selectivity or Defouling Properties Andrew Alderson,*,† John Rasburn,‡ Simon Ameer-Beg,† Peter G. Mullarkey,‡ Walter Perrie,† and Kenneth E. Evans‡ BNFL, Research and Technology, Building 709, Springfields, Preston PR4 0XJ, United Kingdom, and Department of Engineering, University of Exeter, North Park Road, Exeter EX4 4QF, United Kingdom
Micromachined polymeric honeycomb membranes having conventional and re-entrant cell geometries have been fabricated using femtosecond laser ablation. Mechanical properties characterization confirms that the re-entrant membrane is auxetic (possesses negative Poisson’s ratios: νxy ) -1.82 ( 0.05 and νyx ) -0.51 ( 0.01) whereas the conventional membrane possesses positive Poisson’s ratios (νxy ) +0.86 ( 0.06 and νyx ) +0.6 ( 0.1). Comparison with honeycomb theory confirms that the dominant deformation mechanism is flexure of the honeycomb ribs. The auxetic membrane has been challenged with single-sized glass chromatography beads such that the beads were initially resting on the re-entrant cells. Subsequent tensile loading of the membrane showed the auxetic cells opening during deformation, enabling the beads to pass through the membrane. We have modeled the pore-opening properties of both types of membranes, and the observed behavior for the auxetic membrane is consistent with the model. This is a clear proof-of-concept demonstration of the potential of auxetic materials and structures in filter defouling or cleaning operations. This paper, therefore, demonstrates the successful design and fabrication of a micromachined auxetic structure having specifically tailored mechanical properties that show enhanced functional performance over the conventional filter structure. 1. Introduction Auxetic materials exhibit a negative Poisson’s ratio; that is, they expand laterally when stretched and contract laterally when compressed.1 This unusual, counterintuitive behavior is very rare in naturally occurring materials,2 although it has been an accepted consequence of classical elasticity theory for over 150 years.3 Recent interest in this property resulted from a number of synthetic auxetic materials being produced. In nearly all these cases, the auxetic material was formed by altering the internal microstructure of a conventional material, for example, foam4 and microporous polymers,5,6 to produce auxetic behavior. In a rare example of a naturally occurring auxetic material, the auxetic form (R-cristobalite7) can be thought of as a geometric rearrangement of the conventional form, R-quartz. The benefits of having a negative Poisson’s ratio have been investigated and many improvements in mechanical properties have been identified including shear modulus,1 fracture toughness,4 indentation resistance,8 and acoustic response.9 The mechanism for producing a negative Poisson’s ratio is not scale-dependent, so it is possible to envisage structures, rather than materials, exhibiting exactly the same phenomenon. In this situation, other benefits from having an effective negative Poisson’s ratio across the * To whom correspondence should be addressed. Current address: Faculty of Technology, Bolton Institute, Deane Road, Bolton BL3 5AB, U.K. Tel.: +44 1204 903513. Fax: +44 1204 381107. E-mail:
[email protected]. † BNFL. ‡ University of Exeter.
structure have been identified. One is the possibility of producing synclastic curvatures4 in honeycombs for use in composite sandwich panels.10 A second, and the subject of this paper, is the use of auxetic honeycombs and structures in filtration.4,11 One of the main problems of filter systems is the reduction in filtration efficiency because of fouling of the filter itself, by blockage of the barrier pores by particulates. Auxetic materials and structures offer barriers with the potential for cleaning of the membranes to be facilitated in a manner not easily achievable in a conventional barrier material or structure. This entails opening up the pores of the barrier followed by a backflushing (or cross-flow flushing) operation to remove the entrained particulates blocking the barrier. Pore opening may be effected by the application of an external tensile uniaxial load (or displacement) which, for an auxetic material or structure, will cause the pore to open up along and transverse to the direction of the load. For a conventional material, however, the pore would only be extended along the direction of the applied load, the pore undergoing contraction (closing up) in the transverse direction. This is shown schematically in Figure 1 for re-entrant and conventional hexagonal honeycomb networks deforming by hinging of the honeycomb ribs. It has been shown elsewhere that the re-entrant honeycomb structure is auxetic when deformation is due to rib hinging or flexure.12,13 Alternatively, auxetic materials or structures may be used in membrane barriers which compensate for cake fouling in situ. This utilizes a passive pore variation mechanism due to the synclastic curvature4,10 property that results from having a negative Poisson’s ratio. In this scenario, the pressure drop across the barrier
10.1021/ie990572w CCC: $19.00 © 2000 American Chemical Society Published on Web 01/29/2000
Ind. Eng. Chem. Res., Vol. 39, No. 3, 2000 655
Figure 2. Schematic diagram of the Femtosecond laser system. Low-energy 100-fs laser pulses are produced at 790 nm by a modelocked ti-sapphire laser (Coherent Mira 900-F pumped by a Coherent Innova 300 argon ion laser). After being stretched to 300 ps, selected pulses are amplified in a ti-sapphire regenerative amplifier pumped by a kilohertz Nd:YLF laser and recompressed to τp ∼ 170-fs pulselength with pulse energy E ∼ 500 µJ (BMI Alpha 1000).
The auxetic membrane considered here has a polymerbased microstructure, fabricated by femtosecond laser ablation, capable of sufficient distortion for use in filtration. 2. Membrane Fabrication
Figure 1. (a) Undeformed re-entrant honeycomb network; (b) reentrant honeycomb network under an applied stress in the y direction; (c) undeformed conventional honeycomb network; (d) conventional honeycomb network under an applied stress in the y direction. The re-entrant honeycomb is auxetic and demonstrates a widening of the pore dimensions along both the x and y directions under tensile loading. The conventional honeycomb is nonauxetic and undergoes a pore widening along the direction of an applied tensile load accompanied by a pore contraction in the transverse direction.
increases as the cake foulant builds up on the barrier. The increase in pressure drop may cause the barrier to bow, which in turn opens up the pores of the barrier. Thus, an auxetic membrane has an in-built mechanism to compensate for the reduction in effective pore size due to fouling, thereby extending the lifetime of the filter. Such bowing is not possible in conventional membranes.11 Because the auxetic effect is scale-independent, then in principle, auxetic materials and structures can offer enhanced separation barriers for a wide variety of industrial processes, including solid/liquid separations (filtration), gas/gas separations (molecular sieves), and solid/gas separations (air filtration). Here, we examine the use of honeycomb membranes in a proof-of-concept demonstration of the defouling and size-selectivity properties associated with auxetic filters. To use auxetic honeycombs in practical filtration systems, it is necessary to construct honeycombs with cell sizes in the appropriate particulate size range. An appropriate technology for the production of honeycombs having cells in the micrometer size range is the use of laser stereolithography.14 This has been identified as a method for producing auxetic microstructures for improved sensors15 and for micromachine manipulators.16 However, these applications require stiff materials that can only withstand very small strains before failure.
UV laser ablation for micromachining and patterning of polymers using nanosecond excimer lasers is a wellestablished technique.17,18 A strong UV absorption coefficient is desirable (>104 cm-1) and, in the case of weak absorption, significant thermal degradation effects may be observed.19 Femtosecond pulse laser ablation can yield precise materials processing resulting from efficient energy deposition, while simultaneously minimizing heat conduction and thermal damage to the surrounding material. UV, visible, and near-IR femtosecond pulses may be used to microstructure transparent polymers at ultrahigh intensity because multiphoton absorption by chromophores dominates the ablation mechanism.19-21 Micromachined re-entrant and conventional honeycomb polymeric membranes were formed by direct femtosecond laser ablation in air of a sheet of HewlettPackard Color LaserJet Transparency film (part number: HP C2936A). Figure 2 shows a schematic diagram of the femtosecond laser system used for polymer microstructuring. The standard technique of chirped pulse amplification (CPA)22 was used to amplify the lowenergy (∼1-10 nJ) pulses to the millijoule level compatible with the fluences required for laser ablation (∼110 J cm-2 ). Pulses from the 1-kHz titanium sapphire regenerative amplifier at 790 nm (170 fs) and frequencydoubled near-UV pulses at 395 nm (200 fs) were both used effectively in microstructuring. The laser output was directed via a pair of temperature-stabilized XY galvomirrors (General Scanning Inc.) onto a flat field lens (f ) 160 mm) and focused at the substrate surface to ≈100-µm spot diameter. Pulses of 100 µJ corresponding to a fluence of 1.3 J cm-2 were used to mark the cell perimeters at a scan speed of 5 mm s-1. Penetration through the 128-µm thick polymer substrate was achieved after ≈5 overscans/cell (∼100 pulses/spot diameter) corresponding to an ablation rate of ≈1.3 µm/ pulse. In terms of the linear response, the polymer membrane is highly transparent at both wavelengths and absorbs strongly only below 350 nm. Photographs of re-entrant and conventional honeycomb membranes ablated at 790 nm are shown in Figure 3.
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Figure 3. (a) Polymeric re-entrant honeycomb membrane; (b) polymeric conventional honeycomb membrane. Pores are approximately 1 mm in width (along the x direction). The membranes were fabricated by direct femtosecond laser ablation in air, with pulses at 790 nm (170 fs).
The resulting cell dimensions measured from optical micrographs were h ) 0.78 ((0.03) mm, l ) 0.54 ((0.02) mm, t ) 0.086 ((0.006) mm, w ) 0.128 ((0.005) mm, and R ) -23((2)° for the re-entrant membrane and h ) 0.69 ((0.07) mm, l ) 0.56 ((0.02) mm, t ) 0.14 ((0.03) mm, w ) 0.128 ((0.005) mm, R ) +23((2)° for the conventional membrane (see Figure 1 for definition of geometrical parameters). 3. Membrane Characterization Mechanical testing of the honeycomb materials was carried out on a Shimadzu AGS-D 10 kN universal testing machine, using both 10 kN and 500 N load cells. Load data were synchronously logged with displacement data obtained both parallel and transverse to the loading direction using a videoextensometer (Messphysik VideoExtensometer package, Messphysik, Laborgera¨te Ges.M.B.H., Fu¨rstenfeld, Austria). The videoextensometer used differential contrast on a digitized image of the specimen obtained from a CCD (charge coupled detection) video camera (Mintron OS-65D), giving image data of 480 × 640 pixels, to measure displacements. The magnification (ratio of a given distance on the acquired digital image divided by the known actual value of this distance) was 19, giving a resolution of (2 µm corresponding to a strain of (10-4.
Four sets of output were recorded with each test, these being the instantaneous run time of the data points, the load developed on stretching the specimen, and the length and width of the specimen. For the overall deformation of the honeycomb specimens, the latter were measured by an “indirect” means, namely, the movement of thin, bright, adhesive markers placed on the specimen, as detected by the videoextensometry software that is capable of automatically measuring images by changes in contrast. These markers were placed toward the periphery of the honeycomb so that the average deformation behavior of a large number of cells (>20) could be determined. From the raw data, the stress and longitudinal and lateral strains were calculated. A simple demonstration of the feasibility of employing auxetic membranes in filter defouling operations was performed by challenging the re-entrant membrane with glass chromatography beads of diameter 0.42 ((0.02) mm. The beads were on average slightly larger than the “size” of the pores of the membrane so that there was little transmission of the beads from one side of the membrane to the other in the undeformed state. A series of tensile tests were performed and the subsequent bead transmission with strain measured for the re-entrant membrane. This was achieved by counting the number of blocked (or vacant) pores in the field of view at regular time/strain intervals from a video recording of a continuously strained membrane. In total eight particle transmission tests were performed for loading in the x direction and five were performed for loading in the y direction. This number of tests was carried out to determine a representative average of the strain-dependent defouling properties with an acceptable level of experimental uncertainty. In the case of the conventional membrane it was not possible to observe particle transmission for the range of bead sizes available. For beads of diameter 0.68 ((0.05) mm all the beads passed through the conventional membrane in the undeformed state, whereas for beads of diameter 1.03 ((0.03) mm no particle transmission was observed in either the undeformed state or in a strained state up to a tensile strain of ∼1.5%. 4. Results 4.1. Mechanical Properties. Typical longitudinal versus transverse extension data are presented in Figure 4a,b for the re-entrant and conventional (hexagonal) cases, respectively. It is clear, from the width/ extension curves of Figure 4a,b, that the re-entrant honeycomb expands laterally when stretched while the conventional honeycomb contracts laterally. This is also illustrated in Figure 5a,b where the transverse true strain x is plotted versus the longitudinal true strain y applied in the y direction. The slope of this plot gives Poisson’s ratio -νyx (see for example ref 23). An analogous set of data was obtained for the orthogonal direction, giving -νxy. Young’s modulus in each case was determined from the initial slope of the stress-strain curves (Figure 6a,b). The overall results are summarized in Table 1. In all cases the honeycombs remained linear elastic up to strains of about 0.01. The mechanical properties for honeycomb networks for deformation due to rib hinging or flexure have been
Ind. Eng. Chem. Res., Vol. 39, No. 3, 2000 657
Figure 4. (a) Longitudinal vs transverse extension data for reentrant honeycomb membrane; (b) longitudinal vs transverse extension data for conventional honeycomb membrane. Deformation is due to an applied load in the y direction in both cases.
derived elsewhere12,13,24 and are simply quoted here:
νxy ) νyx-1 )
Ex )
cos2 R
(hl + sin R) sin R
K cos R h w + sin R sin2 R l
(
)
h K + sin R l Ey ) w cos3 R
(
)
(1)
(2)
Figure 5. (a) Longitudinal vs transverse true strain data for reentrant honeycomb membrane; (b) longitudinal vs transverse true strain data for conventional honeycomb membrane. Deformation is due to an applied load in the y direction in both cases.
shown that24
Kf ) Esw
(tl)
where Ex and Ey are Young’s moduli due to loading in the x and y directions, respectively, h, l, and R are the geometrical parameters defined in Figure 1, w is the depth of the honeycomb ribs, and K is the force constant for the hinging (Kh) or flexure (Kf) deformation mechanism. From standard beam theory it has been
(4)
where Es is the intrinsic Young’s modulus of the rib material and t is the rib thickness defined in Figure 1. Similarly, if hinging is considered to be due to shearing of the material at the cell wall junction, Kh has been shown to take the form24
Kh ) Gsw (3)
3
(tl)
(5)
where Gs is the shear modulus of the cell wall material. The shear mechanism for hinging is expected to be important for cells where the rib aspect ratio l/t is small. Note that Poisson’s ratios for honeycomb networks are identical when deformation is due to hinging or flexure, depending only on the geometrical parameters of the network (eq 1). The expressions for Young’s moduli associated with the hinging and flexure mechanisms differ only by the form of the respective force constants.
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Table 1. Values for ν and E for Conventional and Auxetic HoneycombssTheory and Experiment Ex, GPa
Ey, GPa
Conventional Membrane 0.6 ( 0.1 0.7 ( 0.1 0.6 ( 0.1
0.23 ( 0.04 0.2 ( 0.2 0.15 ( 0.08
0.130 ( 0.001 0.12 ( 0.08 0.09 ( 0.05
Re-entrant Membrane -0.51 ( 0.01 -0.49 ( 0.05 -0.44 ( 0.04
0.127 ( 0.006 0.10 ( 0.03 0.08 ( 0.02
0.032 ( 0.006 0.024 ( 0.006 0.021 ( 0.005
νxy experimental theory (flexure)12 theory (concurrent)13,24
0.86 ( 0.06 1.4 ( 0.3 1.0 ( 0.2
experimental theory (flexure)12 theory (concurrent)13,24
-1.82 ( 0.05 -2.0 ( 0.2 -1.8 ( 0.1
νyx
Figure 6. (a) Stress-strain data for re-entrant honeycomb membrane; (b) stress-strain data for conventional honeycomb membrane. Deformation is due to an applied load in the y direction in both cases.
The theoretical mechanical properties for the membranes studied here are also presented in Table 1, calculated using eqs 1-4 and assuming deformation is due to rib flexure when Young’s moduli are calculated. The value of Es, the stiffness of the polymer material, was obtained by testing a piece of the unablated polymer material, yielding a value of the intrinsic material modulus Es ) 4.4 ((0.6) GPa. The agreement between experiment and theory for Poisson’s ratio and Young’s modulus data is generally good. With the exception of νxy for the conventional
membrane, variation between the two sets of data is within the calculated uncertainties. 4.2. Particle Transmission. Figure 7 shows stills from a video recording of the particle transmission experiments for loading of the re-entrant membrane in the x direction. The stills cover a strain range of 0 e x e 0.01, which is indiscernible by the eye in Figure 7. The glass beads, which were initially resting on the reentrant cells, began to fall through the membrane pores as tensile strain is applied. It should be noted that in the particle transmission experiments the negative values of Poisson’s ratios were smaller in magnitude than those originally measured (νxy ∼ -1.4 and νyx ∼ -0.18, cf. -1.82 and -0.51, respectively) due to the advent of three broken ribs in the membrane at this stage of the experiment. These imperfections and other small variations in geometrical parameters (of the membrane and the beads) probably account for the fact that not all beads were transmitted at the same instant. The normalized fraction of blocked pores, averaged over all tests, is plotted versus the loading strain in Figure 8 for loading in the x and y directions. The data are normalized to account for different levels of fouling at the start of each test (i.e., n/n0 ) 1 initially for the best fit line to each set of loading data, where n is the number of blocked pores in the field of view and n0 is the initial number of blocked pores in the field of view). The principle of mechanical control of the sieving or filtering action of the membrane is clear from Figures 7 and 8. Furthermore, Figure 8 shows the rate of defouling is dependent in some manner on the value of Poisson’s ratio, as indicated by the different slopes of the data for loading in the x and y directions (slopes ) -30.00 and -9.46, respectively). It was not possible to observe similar particulate defouling behavior with the conventional honeycomb membrane, because of the beads available for testing purposes being either too small (and therefore passing through the membrane in the undeformed state) or too large for significant pore variation to be achieved without the membrane yielding plastically. The case of using a conventional membrane in this scenario is, however, discussed below. 5. Discussion and Conclusions The requirement of a quadratic strain energy function and hence of a symmetric stiffness matrix for an orthotropic linear elastic material implies the following relationship should be satisfied:25
νxyEy )1 νyxEx
(6)
From the experimental data in Table 1, we find (νxyEy)/ (νyxEx) ) 0.8 ((0.2) and 0.9 ((0.2) for the conventional
Ind. Eng. Chem. Res., Vol. 39, No. 3, 2000 659
Figure 7. Video stills of re-entrant membrane defouling when challenged by glass chromatography beads: (a) ∼60% bead coverage on an undeformed membrane; (b) ∼50% bead coverage at a strain of 0.5% in the x direction; (c) ∼40% coverage at a strain of 0.75% in the x direction; (d) ∼30% coverage at a strain of 1% in the x direction.
Figure 8. Averaged normalized fraction of blocked pores vs loading strain for the re-entrant membrane challenged by glass beads. Data for loading in the x and y directions are shown.
and re-entrant honeycomb membranes, respectively. Hence, the experimental data are consistent, within experimental error, with classical anisotropic elasticity theory. Similarly, the requirement that the strain energy of an orthotropic material be positive definite for static equilibrium leads to25
|νxy| e
() Ex Ey
1/2
(7)
which is also satisfied by the experimental data in Table 1 for both membranes. The excellent agreement between theory and experiment clearly demonstrates that honeycomb structures can be designed and fabricated using femtosecond laser ablation with specifically tailored mechanical properties. Auxetic and nonauxetic structures can be made from the same material by modifying the geometry of the structure, for example, by changing R or h/l. 5.1. Deformation Mechanisms and Force Constants. In a comparison of the experimental mechanical properties with theory, it was assumed that deformation of the micromachined structures was via flexure of the honeycomb ribs. This is vindicated by comparing the force constants for rib flexure and hinging given in eqs 4 and 5, which leads to
Kf Es t 2 t2 ) ) 2(1 + νs) K h Gs l l
()
()
(8)
where νs is Poisson’s ratio of the rib material which we
assume to be isotropic, allowing us to use the standard relation from classical elasticity theory of isotropic materials: Es ) 2Gs(1 + νs). Experimentally, we determined νs ) 0.56 ((0.10), which is consistent with the maximum positive value of +1/2 allowed by classical elasticity theory for isotropic materials, or may be indicative of some degree of anisotropy in the film due to the drawing process during manufacture. Hence, employing the values of t/l established above, we find for the conventional honeycomb structure Kf/Kh ) 0.18 ((0.08), and for the re-entrant honeycomb structure Kf/ Kh ) 0.08 ((0.01). The lower the value of the force constant for a given deformation mechanism, the more significant that mechanism becomes,13,24 and so we see that flexure dominates over hinging for both types of membranes in this case. A third possible deformation mechanism exists for these membranes, corresponding to stretching of the ribs in response to an applied load. In this case the force constant is given by13,24
Ks ) Esw
(tl)
(9)
leading to Kf/Ks) 0.06 ((0.02) and 0.026 ((0.003) for the conventional and re-entrant honeycomb membranes, respectively. Hence, flexure again dominates. A model has been developed where all three deformation mechanisms act concurrently.13,24 An example of the concurrent model expression is given here for νxy:
[
sin R cos2 R νxy )
(
)[(
h + sin R l
1 1 1 + Kf Kh Ks
)
]
]
1 1 cos2 R + sin2 R + Kf Kh Ks
(10)
To check the validity of the force constant expressions (eqs 4, 5, and 9), the concurrent model calculations for the membranes considered here are also presented in Table 1. Again, good agreement is achieved with experiment, as expected because flexure dominates and we have already seen that the experimental data are in good agreement with the flexure model. The concurrent model predictions are slightly different from those of the flexure model (because of the influence of the hinging and stretching mechanisms), but the experimental errors are too large to enable a definitive statement as to which model best represents the experimental data. Note, though, that the experimental νxy value for the conventional membrane agrees within error with the concurrent model predictions. This was found not to be true using the flexure model alone.
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5.2. Strain-Dependent Deformation. The straindependent deformation and particle transmission of the membranes considered here have also been investigated; see Figures 5-8. We have also developed simple analytical expressions for the maximum inscribed sphere for the pores of both membranes. It is, therefore, possible to model the particle transmission with strain for both membranes, assuming deformation is due to rib hinging (i.e., R varying); see below. However, it has been noted above that rib flexure appears to dominate the deformation of the membranes. Hence, in the following we consider the strains in the x and y directions for deformation to be due to rib flexure and hinging to (i) facilitate comparison with the experimental strain data (Figure 5a and 5b) and (ii) compare the strain data for rib flexure and rib hinging to assess the viability of subsequently using the hinging model to model the experimental strain-dependent particle transmission data. True strains are evaluated from
( )
x ) ln
X ) ln(1 + ex) X0
(11)
where X and X0 are the deformed and undeformed unitcell lengths in the x direction, respectively, and ex is the engineering strain in the x direction. Similar expressions exist for strain in the y direction. For example, in the case of the flexure model, ex is given by12
ex )
δ tan R l
(12)
where δ is the deflection of the rib undergoing flexure. In the flexure model, therefore, strains are calculated by varying δ, while maintaining l and R constant. This is applicable to low strains where beam theory26 is valid, as is the case for the experimental data presented in Figures 5 and 6. In the case of the hinging model x is given by
(
cos R x ) ln cos R0
)
(13)
where R and R0 are the deformed and undeformed honeycomb angle, respectively. In the hinging model strains are therefore calculated by varying R while maintaining all other geometrical parameters constant. In Figure 9a we show the comparison of the experimental x versus y data with the flexure and hinging models for the re-entrant membrane. The two models are almost coincidental throughout the entire strain range (0 e y e 0.010). Furthermore, the agreement with the experimental data is also good, with the models having only a slightly lower gradient than that shown by the experimental data. This is of course reflected in the excellent agreement between the theoretical and experimental values of νyx for the re-entrant honeycomb membrane (see Table 1). The experimental and theoretical (flexure and hinging models) x versus y data for the conventional membrane are shown in Figure 9b. Again, the hinging and flexure models give almost identical results throughout the strain range considered in Figure 9b. The experimental trends are also reasonably well reproduced, although there is slightly more discrepancy in the model and experimental slopes for the conventional membrane
Figure 9. (a) Theoretical (hinging and flexure models) and experimental x vs y data for the re-entrant honeycomb membrane; (b) theoretical (hinging and flexure models) and experimental x vs y data for the conventional honeycomb membrane. Hinging model ) solid line; flexure model ) dashed line.
than for the re-entrant membrane (Figure 9a). However, we note that the calculated uncertainties in the slope of the model curves are of the order of 15-20% (see flexure model νxy and νyx data in Table 1). Hence, the experimental strain data agree within the calculated uncertainties with the flexure model data. It also appears reasonable to use the rib hinging model as a first approximation to model other strain-dependent properties (e.g., particle transmission) for these membranes. 5.3. Strain-Dependent Particle Transmission. In the simple particle transmission tests performed here we have demonstrated the potential of auxetic materials or structures in filtration applications, specifically in the cleaning of fouled membranes. This has been achieved by the application of a uniaxial load or displacement to open up the pores in a manner unique to auxetic materials or structures, thereby allowing the passage of glass beads from one side of the re-entrant membrane to the other. This was not observed in the comparative tests on the conventional membrane. This may be due to the fact that the conventional honeycomb cells simply
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Figure 10. The maximum radius of a sphere capable of passing through the pores of the re-entrant and conventional honeycombs studied here plotted as a function of loading strain in the y direction. The values are normalized to the radius of the inscribed sphere at the undeformed geometry (y ) 0). Geometrical values determined experimentally (see text) were used in the calculations. The pore walls constraining the maximum sphere size are shown schematically in the figure.
contract in the lateral dimension when stretched longitudinally (see Figure 1), thus reducing the size of the sphere able to pass through the membrane. Alternatively, it may be due to the beads employed in the test not having a diameter with a close enough match to the conventional pore size in this case. To clarify this issue further, we have modeled various situations involving the admission of a uniform sphere through six-sided re-entrant and conventional cells. Although we have ascertained that rib flexure is the dominant mechanism, we have seen that the strain and Poisson’s ratio calculations are almost identical between the flexure and hinging models and so the maximum bead size analysis assumes deformation is due to rib hinging for simplicity. In this analysis, the maximum inscribed sphere radius R when the vertical pore ribs constrain the sphere (see Figure 10) is given by
R ) l cos R -
t 2
(14)
Equation 14 is valid for both the conventional and reentrant honeycomb geometry. In the case where the diagonal ribs constrain the bead, for the conventional membrane we have
R)
t h cos R + l sin R cos R 2 2
(15)
and for the re-entrant membrane
R)
t π R h tan + 2 4 2 2
(
)
(16)
Figure 10 shows the variation in the radius of the sphere able to pass through the pores of conventional and reentrant membranes having the geometrical parameters established above for the membranes considered in this paper. The radius is plotted versus the applied strain
in the y direction and has been normalized to the radius of the sphere able to pass through the pore in the undeformed membrane in each case. In effect, the normalization enables a comparison between re-entrant and conventional membranes having the same initial effective pore size. In the calculations we have assumed deformation to be due to rib hinging. For the re-entrant membrane, it is clear that the size of the particle that can pass through the membrane can be increased significantly by applying a tensile load or strain in the y direction. The size of the particle is in this case constrained by the diagonal ribs (length l). The case where the particle is constrained by the vertical ribs (length h) is possible but requires h/l > 2. From eq 16 the geometrical parameters measured for the re-entrant membrane (see above) yield a maximum bead diameter ()2R) for transmission through the membrane of 0.43 ( 0.03 mm. This corresponds well with the known value of 0.42 ((0.02) mm for the beads employed in the test, enabling this simple bead and membrane system to demonstrate proof of principle for auxetic materials or structures in filter defouling or size selectivity operations. For the conventional membrane the size of particle that the pore can accommodate is at a maximum (within error) when the membrane is in the undeformed state for the membrane geometry considered here. Equation 15 yields a value for the maximum inscribed bead diameter for the pores of the conventional membrane of 0.89 ( 0.05 mm, which explains why the beads of diameter 0.68 ((0.05) mm passed through the undeformed membrane whereas those of diameter 1.03 ((0.03) mm did not. Application of a tensile or compressive load or strain in either direction for the conventional membrane only leads to a decrease in the size of bead able to pass through the membrane, which explains why the 1.03-mm diameter beads were not transmitted when the conventional membrane was deformed. The conventional membrane is, therefore, unsuitable for the type of defouling operation afforded by the auxetic membrane. The maximum particle size a conventional honeycomb membrane (with h/l < 2) can accommodate occurs at the geometry at which the ribs constraining the particle change from being the diagonal ones to the vertical ones (see Figure 10). (For a conventional honeycomb membrane with h/l g 2 the vertical ribs always constrain the bead size, and so the angle corresponding to the maximum particle size occurs at R ) 0°.) Clearly, it is possible to design conventional honeycombs (with h/l < 2) with an undeformed pore “size” away from the maximum, which would allow some potential defouling operation due to increasing the pore size. However, away from the “regular hexagon” geometry other effects due to, for example, pressure drop and particle flux through the membrane may need to be taken into account. The rate of particulate defouling with loading strain has been found to be related in some manner to Poisson’s ratio for the particular direction of loading (see Figure 8). Consider the case of loading in the y direction. Here, the rate of defouling with respect to the loading strain (y) is related to the rate of defouling with respect to the transverse strain (x) by
(
) (
d(n/n0) dy
)-
σy
)
d(n/n0) dx
νyx
σy
(17)
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where the subscript σy indicates that the strain-dependent rates of defouling are due to an applied load in the y direction, and we have used the definition for Poisson’s ratio of
dx νyx ) dy
(18)
which is applicable to nonlinear as well as linear deformation.23,27 If we assume that the rate of defouling with strain in a given direction is independent of the direction of loading, for example,
(
) (
d(n/n0) dx
)
σx
)
d(n/n0) dx
(19)
σy
then, from eqs 17 and 19, the ratio of the slopes of the two curves in Figure 8 is given by
( (
) )
d(n/n0) dx
σx
d(n/n0) dy
σy
)-
1 νyx
(20)
Similarly, it can also be shown that the ratio of the slopes is related to νxy by
( (
) )
d(n/n0) dx d(n/n0) dy
σx
) -νxy
(21)
σy
In the case of the flexure or hinging models, eq 1 is valid and eqs 20 and 21 are equivalent. However, for models or structures where νxy * νyx-1, the assumption that the rate of defouling with strain in a given direction is independent of the direction of loading may not be valid. This is the case for the concurrent model and membranes discussed in this paper. Furthermore, the membrane used in the particle-transmission experiment contained a small number of damaged ribs, further invalidating the assumption in eq 19. In such cases, eqs 20 and 21 give different values, which we take to be the limiting cases. Taking the average of eqs 20 and 21 for Poisson’s ratios associated with the membrane during the particle transmission experiment (νxy ) -1.4 and νyx ) -0.18) gives a predicted ratio of the slopes for Figure 8 of 3.5, which is in good agreement with the observed value of 3.2. Hence, the magnitude of Poisson’s ratio is a measure of the sensitivity of the membrane to changes in the pore size. To check for consistency in the particle transmission data for loading in the x and y directions, plotted versus the loading strain (i.e., x and y, respectively) in Figure 8, it is instructive to plot the data versus the strain in one direction only. To a first approximation each set of data in Figure 8 can be fitted by a straight line which in the case of the x-loaded data is given by
(
)
d(n/n0) n ) n0 dx
σx
x + 1
(22)
Figure 11. Averaged normalized fraction of blocked pores vs strain in the x direction for the re-entrant membrane challenged by glass beads. Data for loading in the x (crosses) and y (circles) directions are shown. Also shown is the calculated curve assuming a normal cumulative distribution of glass bead sizes having a mean value of R/l ) 0.415 and a standard deviation of 0.019. All membrane geometrical parameters were as observed experimentally.
where [d(n/n0)/dx)]σx is the gradient of the straight line ()-30.00 in Figure 8). Similarly, for loading in the y direction
(
)
d(n/n0) n ) n0 dy
y + 1
(23)
σy
Equations 20 and 21 provide conversion between the gradients of the straight lines in Figure 8, defined by eqs 22 and 23. Taking the average of the rate of defouling with respect to strain for loading in the x direction given by eqs 20 and 21 (due to the assumption that the rate of defouling with strain in a given direction is independent of the direction of loading being invalid in this case; see above), we have
〈( ) 〉 d(n/n0) dx
σx
(
1 + νxy νyx )2
)
( ) d(n/n0) dy
(24)
σy
Rearranging eq 22 and substituting eqs 23 and 24 yields
x ) -
2y 1 + νxy νyx
(
)
(25)
Equation 25 provides the necessary conversion factor to enable the particle transmission data due to loading in the y direction to be plotted versus the same strain as the data due to loading in the x direction. (Note that if the assumption of the rate of defouling being independent of the loading direction was valid, then νxy ) νyx-1 and so eq 25 would simplify to x ) -νyxy.) This is illustrated in Figure 11 where the experimental normalized fraction of blockages (n/n0) are plotted versus x for uniaxial loading along both the x and y directions. In Figure 11, the data due to loading along the x axis (crosses) are as given in Figure 8. The data due to loading along the y axis (circles) have also been included by using eq 25 to convert the strain data. Using
Ind. Eng. Chem. Res., Vol. 39, No. 3, 2000 663
this method, it is readily seen that the data for loading along the x and y directions are consistent, because they now overlap once the effects of Poisson’s ratios and loading direction have been taken into account. For a bead/membrane system consisting of beads of exactly identical shape and size, and a membrane having exactly identical pore structure and rib geometry throughout the membrane, it might be expected that complete defouling would occur at the instant the pore size exceeds the bead diameter. In reality, of course, the beads will have a distribution of sizes, and there will be some variation in the pore geometry across the membrane, leading to particle transmission occurring across a range of strains, as observed in Figure 8. Therefore, we have modeled the particle transmission through the re-entrant membrane assuming a normal distribution of particle sizes going through pores of a particular size governed by eq 16 for the membrane geometry considered in this paper. In the calculations, therefore, those particles in the distribution having a radius less than that determined from eq 16 are considered to be transmitted through the membrane. Particle transmission was calculated for varying R (i.e., assuming membrane deformation is due to rib hinging). The normalized fraction of blockages calculated in this way are also plotted in Figure 11 (solid curve), using eq 13 to convert the variations in R to x. The model curve is normalized to n/n0 ) 1 at the undeformed geometry to ensure consistency with the experimental data. A normal cumulative distribution of particles having a mean value of R/l ) 0.415 and a standard deviation of 0.019, with all other membrane geometrical parameters as observed experimentally, gives the best fit to the experimental data in Figure 11. The mean value of R/l employed in the calculations is, therefore, in excellent agreement with the value of R/l ) 0.40 ( 0.04 calculated using eq 16 from the experimentally determined geometrical parameters for the undeformed membrane, which we have already seen possesses a pore size which corresponds well with the size of beads used in the test. In this paper the potential of employing auxetic materials in filter defouling operations has been considered under uniaxial loading conditions. Of course, biaxial loading operations may be possible in certain filtration applications. Tensile loading in either the x or y direction leads to an increase in pore dimensions of an auxetic membrane, and so the effects are additive in the biaxial loading case, leading to further enhanced pore-opening possibilities. However, for a conventional membrane the effects due to tensile loading in each direction oppose each other from a pore-opening point of view (i.e., one tends to open the pore in the y direction while closing the pore in the x direction and vice versa). This is demonstrated by considering eq 20 or 21 where positive Poisson’s ratio values lead to a negative ratio of the slopes of the defouling with strain curves for loading in the x and y directions, indicating that defouling may be possible by loading in one direction but not the other. Hence, once again we expect auxetic materials or structures to offer improvements in filter defouling operations. An alternative mechanism for effecting pore variation may be as a result of the passive response of a membrane subject to a pressure-drop build-up across the membrane as a result of membrane fouling. This may be considered equivalent to the application of
bending moments leading to curvature (“bowing”) of the membrane in the direction of flow. It is known that materials and structures possessing a negative in-plane Poisson’s ratio undergo synclastic curvature when subject to a bending moment, whereas those possessing a positive in-plane Poisson’s ratio undergo anticlastic curvature.4,10 In other words, auxetic materials and structures can be deformed into a doubly curved “dome” shape more easily than nonauxetic materials and structures. Such deformation will lead to an increase in pore size which, as we have demonstrated in this paper, may be used for membrane defouling or for reducing a pressure increase during fouling. However, it should be noted that the bowing action due to pressure build-up could also introduce detrimental effects such as the release of initially trapped particulates into the downstream. The transmission of particles through the filter due to bowing may also hinder the establishment of the more effective filtration phase where the initial deposit acts as part of the filtering media to trap finer particles. In the case of thick filtering media then the bowing mechanism may not be applicable. In this case the pressure build-up across the filter would tend to close the pores which would oppose the effect of tensile lateral forces that may be applied to open up the pores for defouling purposes. Clearly, the anticipated forces due to pressure build-up need to be considered in the design and selection of the appropriate filter material and control system to ensure satisfactory defouling occurs in reality. The pore-opening properties characteristic of auxetic materials and structures may also be of benefit in applications where, for example, tuneable pore size variations or oscillations are required, or where dynamic alterations of pore size are needed. Auxetic materials and structures have potential in smart filtration applications requiring both active and passive control of the pore size. Future work will involve the use of alternative fabrication techniques to enable auxetic structures having smaller pore sizes to be fabricated and tested. For example, the LIGA (Lithographie, Galvanoformung, Abformung) micromachining technique, which employs lithography, electroplating, and moulding processes to manufacture finely defined microstructures for use in microelectronic and microfiltration components, for example, has been used to fabricate conventional honeycomb structures having pores up to 2 orders of magnitude smaller than the membranes fabricated and tested here. Hence, this technique enables the possibility of auxetic membranes for use in practical air filtration systems to be considered. Scaled-down auxetic honeycombs may also find use in biomedical applications such as ultrafiltration and drug-release membranes where the filtering or release of a clearly defined dose of similarly sized cells/ molecules is required. The fabrication, characterization, and testing of auxetic forms of more realistic filter materials than the membrane studied here for the proof-of-concept demonstration are also now required. Auxetic forms of polymeric foams used in air filtration are known,4 which are obvious choices for study in this respect. Auxetic foams could also be used to support auxetic membranes, similar to the one studied here, for large-scale filtration operations. In this case the support would be required to be highly deformable and have a larger effective pore
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size and matching in-plane Poisson’s ratios with respect to the membrane for the auxetic membrane to exhibit the full benefits in filter defouling and size selectivity demonstrated here. Auxetic metallic foams4 should also be useful in this respect, particularly in applications where increased support stiffness is required. Alternatively, deeper honeycomb monoliths having larger effective pore sizes could be used to support honeycomb membranes. Because of the scale-independent nature of the auxetic effect, the analytical models described in this paper would enable a close match of the in-plane mechanical properties of the membrane and the monolith to be achieved in this scenario. 6. Summary In summary, we have used femtosecond laser ablation to fabricate micromachined auxetic and nonauxetic polymeric membrane structures. The structures have mechanical properties in good agreement with the flexure model for the deformation of honeycombs, and also in good agreement with a more detailed model in which deformation due to rib hinging and stretching are also incorporated. The experimental data are consistent with both models because of flexure being the dominant mechanism in the real structures. Simple tests have been performed, demonstrating that auxetic membranes have beneficial properties over conventional membranes in filter defouling and controlled pore size applications. The strain-dependent defouling is seen to be dependent on the value of Poisson’s ratios of the membrane. Acknowledgment This work has been funded by BNFL and the Engineering and Physical Sciences Research Council of the United Kingdom. Nomenclature ex ) engineering strain along the x direction Es ) intrinsic Young’s modulus of honeycomb rib material, GPa Ex ) Young’s modulus in the x direction, GPa Ey ) Young’s modulus in the y direction, GPa f ) flat field lens focal length, mm Gs ) intrinsic shear modulus of honeycomb rib material, GPa h ) vertical honeycomb rib length, mm Kf ) flexure force constant, N/m Kh ) hinging force constant, N/m Ks ) stretching force constant, N/m l ) diagonal honeycomb rib length, mm n ) number of blocked pores in the field of view n0 ) initial number of blocked pores in the field of view R ) radius of inscribed sphere, mm t ) honeycomb rib thickness, mm w ) honeycomb rib depth, mm X ) deformed unit-cell length along the x direction, mm X0 ) undeformed unit-cell length along the x direction, mm R ) honeycomb angle, deg R0 ) undeformed honeycomb angle, deg δ ) deflection of honeycomb rib due to flexure, mm x ) true strain along the x direction y ) true strain along the y direction νs ) intrinsic Poisson’s ratio of honeycomb rib material
νxy ) Poisson’s ratio for uniaxial loading along the x direction νyx ) Poisson’s ratio for uniaxial loading along the y direction σx ) applied stress along the x direction σy ) applied stress along the y direction τp ) laser pulse length, fs
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Received for review August 2, 1999 Revised manuscript received November 23, 1999 Accepted November 24, 1999 IE990572W
