## Abstract

Most materials shrink laterally like a rubber band when stretched, so their Poisson's ratios are positive. Likewise, most materials contract in all directions when hydrostatically compressed and decrease density when stretched, so they have positive linear compressibilities. We found that the in-plane Poisson's ratio of carbon nanotube sheets (buckypaper) can be tuned from positive to negative by mixing single-walled and multiwalled nanotubes. Density-normalized sheet toughness, strength, and modulus were substantially increased by this mixing. A simple model predicts the sign and magnitude of Poisson's ratio for buckypaper from the relative ease of nanofiber bending and stretch, and explains why the Poisson's ratios of ordinary writing paper are positive and much larger. Theory also explains why the negative in-plane Poisson's ratio is associated with a large positive Poisson's ratio for the sheet thickness, and predicts that hydrostatic compression can produce biaxial sheet expansion. This tunability of Poisson's ratio can be exploited in the design of sheet-derived composites, artificial muscles, gaskets, and chemical and mechanical sensors.

When stretched, most materials contract in both lateral dimensions to decrease stretch-induced volume change. The ratio of percent lateral contraction to percent applied tensile elongation is the Poisson's ratio. Some rubbers have Poisson's ratios of about 0.5 for both lateral directions, so their volume does not appreciably change upon stretching. In very rare materials the sum of Poisson's ratios for lateral dimension changes exceeds unity, so they increase density when stretched and, inversely, expand in at least one direction when hydrostatically compressed (*1*). If a lateral dimension expands during stretching, the associated Poisson's ratio is negative and the material is called auxetic (*2*). Recent interest in this counterintuitive behavior originated from pioneering discoveries that partially collapsed foams and honeycombs (*2*, *3*), fibrillar polymers (*4*), and polymer composites (*5*) can be auxetic.

Poisson's ratio was unknowingly used 2000 years ago in the empirical selection of cork for wine bottle stoppers. Cork stoppers have a near-zero Poisson's ratio for radial directions when subjected to orthogonal uniaxial stress (*6*). A positive Poisson's ratio makes a stopper difficult to insert but easy to remove, and the reverse occurs for a negative Poisson's ratio.

Carbon nanotube sheets (buckypaper) were fabricated (*7*, *8*) by filtration of aqueous dispersions of single-walled nanotubes (SWNTs) (*9*) and multiwalled carbon nanotubes (MWNTs) (*10*) produced by chemical vapor deposition, a technique reminiscent of ancient methods for making writing paper by drying a fiber slurry. The SWNTs are seamless cylinders of graphite about 1 nm in diameter, and the MWNTs consist of about nine concentric SWNT shells having an outer diameter of about 12 nm (*8*). Scanning electron microscopy shows that bundles of the SWNTs (diameter about 20 nm) and largely unbundled MWNTs are intimately commingled in sheets comprising both nanotube types.

The Poisson's ratios of SWNT and MWNT buckypaper were initially determined by using an optical microscope to measure the change in sheet width as a function of stretch in the sheet length direction. More accurate Poisson's ratio measurements resulted from recording movies of the movements of reflective particles on the sheet surface during constant-rate deformation. Image correlation software was used to derive the relative displacements of about 7000 particles per movie frame and corresponding strains in lateral and tensile directions (*8*). The thickness-direction Poisson's ratio was obtained from scanning electron micrographs showing sheet thickness versus applied in-plane tensile strain (*8*).

The in-plane Poisson's ratio was positive (about 0.06) and little changed until the MWNT content reached 73 weight percent (wt %). Further addition of MWNTs decreased the Poisson's ratio to –0.20. A highly nonlinear dependence of Young's modulus, strength, and toughness on MWNT content was also observed, with maximum density-normalized value between compositional extremes, although electronic conductivity and density were approximately a linear function of MWNT content (Fig. 1, A and B, and fig. S1). Perhaps important for applications, the characterized buckypaper comprising both SWNTs and MWNTs had up to 1.6 times the maximum strength-to-weight ratio, 1.4 times the maximum modulus-to-weight ratio, and 2.4 times the maximum toughness of sheets comprising only SWNTs or MWNTs (Fig. 1, A and B, and fig. S1). Large positive Poisson's ratios were observed for the thickness direction (0.33 and 0.75 for SWNT and MWNT sheets, respectively).

Annealing the nanotube sheets at 1000°C in argon for 15 min did not eliminate the sharp change from positive to negative in-plane Poisson's ratio at 77 to 80 wt % MWNTs, indicating that any residual surfactant in the buckypaper is unimportant. Although hysteretic stress-strain curves were seen upon high-strain unloading and reloading, with hysteresis loop width decreasing with decreasing MWNT content and decreasing strain increment for the loop, the Poisson's ratio was essentially constant over these curves.

The negative in-plane Poisson's ratio of MWNT buckypaper sharply contrasts with that reported previously for ordinary paper, made either commercially or by an ancient method like the one we used, where the experimentally observed in-plane Poisson's ratios are large and positive and theoretical calculations predict the possibility of either positive or negative in-plane Poisson's ratios (*11*–*20*). Although entropy and associated thermally driven deviations from in-plane alignment may produce negative in-plane Poisson's ratios for nanofiber biological membranes and related structures (*17*, *21*), persistence lengths of about 0.1 mm at room temperature for even unbundled SWNTs (*18*) mean that similar entropic effects are not important for determining the Poisson's ratios of buckypaper.

A simple model was developed that predicts the observed negative and positive in-plane Poisson's ratios for sheets of different composition, as well as a much larger positive Poisson's ratio in the thickness direction (Fig. 1C and fig. S2). This model captures key structural features of the carbon nanotube sheets: (i) isotropic in-plane mechanical properties; (ii) nanotubes preferentially oriented in the sheet plane, but positively and negatively deviating from this plane by an average angle γ; and (iii) freedom to undergo stress-induced elongation as a result of straightening meandering nanotubes and changing the angle between intersecting nanotubes. This model can also be applied to other fiber networks having similar structure to buckypaper. Although periodic network models have been previously deployed for fiber sheets (*13*), the model structures have high in-plane anisotropy, so uncertainty in the sign of the Poisson's ratio results from the need to average in-plane properties to obtain those for isotropic sheets.

In our model, each meandering nanotube (or nanotube bundle) is represented by a zigzag chain parallel to the sheet plane (with angle of ±γ between the struts and the basal plane). Zigzag chains in one nanotube sheet layer connect via noncovalent interactions with those in adjacent layers at the extremes of the zigs and zags, where torsion about the contact enables change in the angle of intersection between nanotubes. The ratio of the force constant for angle bend to the force constant for changing the inter-nanotube torsional angle is *R*. These force constants for angle bending at zigzags (*k*_{θ}) and torsion between contacting nanotubes (*k*_{T}) are effective values, arising in the intractable real structure from the energy needed to straighten meandering nanotubes and change the angles between intersecting nanotubes (*22*).

Expressing the total energy needed for a given tensile strain in terms of angle bend and torsional angle changes, and minimizing this energy subject to the constraint that all layers have the same tensile-direction and width-direction strains (*8*), provides in-plane (ν_{1}) and sheet thickness direction (ν_{3}) Poisson's ratios of ν_{1} = [*R* – 3 sin^{2} γ]/[3(*R* + sin^{2} γ)] and ν_{3} = [*R* + 3 sin^{2} γ]/[3 tan^{2} γ(*R* + sin^{2} γ)].

These equations predict the observed ν_{1} and ν_{3} (ν_{1} = –0.20 and ν_{3} = 0.75 for MWNT sheets; ν_{1} = 0.06 and ν_{3} = 0.33 for SWNT sheets) for MWNT sheets with γ = 42° and *R* = 0.67 and for SWNT sheets with γ = 50° and *R* = 2.28. The predicted γ values are consistent with average angles from x-ray diffraction of 41.7° for MWNT sheets and 45.0° for SWNT sheets, as well as previous diffraction measurements for similar SWNT sheets (*23*).

The above equation for the in-plane Poisson's ratio can be expressed as ν_{1} = (1 – β)/(3 + β), where β = (3 sin^{2} γ)/*R* = 3(*k*_{T}/*k*_{θ})sin^{2} γ. If the approximation that nanotube struts are rigid is eliminated (*8*), we obtain the same dependence of ν_{1} on β, but with β = 3*k*_{T}/*k*_{SB}, where *k*_{SB} is the force constant for elongation of the nanotube (or nanotube bundles) via a combination of nanotube straightening (changing γ in the present model) and strut deformations. Using a different model, which includes a host of structural and force constant parameters in β, the above dependence of ν_{1} on β has been predicted for sheets of ordinary paper (*12*).

Why do cellulose-based paper (*16*), SWNT paper, and MWNT paper provide such different measured in-plane Poisson's ratios (about 0.30, 0.06, and –0.20), even when they are similarly handmade by removing water from sedimented fiber mats? Insights result from first considering the basic types of deformation that lead to extreme positive and negative in-plane Poisson's ratios. Stretch-induced increases in nanotube length without change in the angle between intersecting nanotubes provides the most negative Poisson's ratio (–1, corresponding to an infinite *k*_{T}/*k*_{θ} or *k*_{T}/*k*_{SB} for the above simplified models). Stretch-induced decreases in the angles between intersecting nanotubes increase Poisson's ratio, producing a Poisson's ratio of 1/3 when β vanishes. To visualize this, note that two neighboring nanotube layers in the Fig. 1C model are coupled like the struts of a wine rack. If rotation between struts dominates, as in an ordinary wine rack, the Poisson's ratio is positive. If torsional rotation of struts is blocked and the struts are stretchable but not bendable, increases in strut length produce a negative Poisson's ratio.

To understand the nanoscale origin of the major differences in Poisson's ratio for nanotube fiber mats, consider that beam bending in response to tensile stress has the same effect for increasing Poisson's ratio as does torsional rotation. In the wine rack analog, a positive Poisson's ratio would result if the hinges are welded to struts to prohibit torsional rotation and the struts are much easier to bend than to stretch. Nanotube beam bending in response to a tensile stress within the sheet plane changes the effective angle between intersecting nanotubes and produces a corresponding increase in Poisson's ratio, similar to the response when there are changes in torsional angle for the model of Fig. 1C. If fiber beam bending is the predominant deformation that changes the effective angle between intersecting fibers, and if fiber deviation from in-plane orientation is neglected, then β = 3*k*_{B}/*k*_{SB}, where the force constants for fiber bending and tensile fiber elongation are *k*_{B} and *k*_{SB}, respectively (*8*). Using this approximation, Perkins (*12*) predicted that the in-plane Poisson's ratio of paper sheets will be positive, with a ν_{1} between 0.259 and 1/3, which is consistent with observations.

Why then do we see a negative Poisson's ratio of –0.20 for MWNT sheets and a much smaller positive Poisson's ratio of 0.06 for SWNT sheets? We provide evidence that these large changes in Poisson's ratio arise from changes in the ratio of beam bending to beam stretch force constants. The tubular nanofiber form and the degree of coupling between SWNT walls are important. The MWNTs we used have an outer diameter of about 12 nm, contain about nine walls, and are largely unbundled in the sheets; the SWNTs have an average diameter of about 1.0 nm (*24*, *25*) and a wide distribution in bundle diameters, with an average around 20 nm.

Consider *k*_{B} and *k*_{S} for a perfectly straight SWNT having strut length *L*: *k*_{B} = 3π*C*(*r*/*L*)^{3} and *k*_{S} = 2π*Cr*/*L*, where *C* is the product of the graphene sheet Young's modulus and sheet thickness, and *r* is the radius of the SWNT (*18*, *26*). The largest geometrically possible value of *r*/*L* is sin(60°)/2, which corresponds to the physically unreasonable case where each layer within the nanotube sheet comprises straight nanotubes that are close-packed within the layer. From β = 3*k*_{B}/*k*_{S} = (9/2)(*r*/*L*)^{2} for the perfect SWNT and ν_{1} = (1 – β)/(3 + β), this hypothetical buckypaper of perfectly straight, infinitely long, unbundled SWNTs cannot have a Poisson's ratio below 0.04. Because the value of β for buckypaper-like sheets comprising long circular solid fibers is the same as for a SWNT when the effective Young's modulus for bending equals that for tension, the predicted ν_{1} is also 0.04 or higher—and likely much higher, because buckypaper-like sheets of ordinary paper do not have fibers that are close-packed in a plane.

Low mechanical coupling between the component SWNTs nested within MWNTs and between SWNTs in bundles within the buckypaper reduces *k*_{B} and *k*_{S}. Low interwall coupling between outer and inner walls is indicated for MWNTs by the sword-in-sheath type of mechanical failure and the ease of pulling the inner nanotubes from a MWNT (*27*, *28*). Because the following calculations treat only the in-plane Poisson's ratio, we here approximate the structure as comprising successive layers of bundled SWNTs or largely unbundled MWNTs (like that shown in Fig. 1C, but with γ = 0°), with stretch force constant *k*_{SB} to accommodate stretch force reduction due to nanotube meandering. This structure provides a volume per strut of *D'L*^{2} sin(120°) and a calculated density of ρ_{calc} = *W*_{L}/[*D'L* sin(120°)], where *W*_{L} is the strut weight per unit length and *D* 'is the sum of the covalent diameter of the nanotube and the 0.34-nm van der Waals diameter of carbon (*8*). By equating calculated sheet densities to the observed sheet densities, interjunction lengths of 54.3 nm and 39.5 nm are calculated for the MWNT and SWNT sheets, respectively. Although these distances seem shorter than suggested by the micrographs of Fig. 2, note that the micrographs are for the sheet surface (the face originally in contact with the filter membrane) and do not provide the junction density and corresponding *L* in the buckypaper interior.

To predict the Poisson's ratio of buckypaper, we calculate the elongation force constant *k*_{SB} (*8*) from the observed Young's modulus *Y* according to *k*_{SB} = 2*D'Y* sin(120°)/(1 – ν_{1}). Although this little affects the results, ν_{1} in this equation is the self-consistently calculated value instead of the measured value. The *k*_{B} for the MWNTs is the sum of bending force constants for all component SWNTs nested within the MWNTs, and the *k*_{B} for SWNT bundles is derived from the measured average Young's modulus for bending (*Y*_{B}) 20-nm-diameter SWNT bundles (50 GPa) (*29*), using the force constant for bending a solid cylindrical rod, *k*_{B} = 3π*R*^{4}*Y*_{B}/(4*L*^{3}) (*26*). The predicted Poisson's ratios are –0.17 for MWNT buckypaper (versus the observed –0.20) and 0.17 for SWNT buckypaper (versus the observed 0.06). Increasing *Y*_{B} to 81 GPa decreases the calculated ν_{1} for SWNT buckypaper to the observed value, and this *Y*_{B} is within the range of experimental uncertainty for *Y*_{B} (*29*).

These results indicate that large negative Poisson's ratios can be achieved by using large-diameter MWNTs having as many interior walls as possible. Although all nanotube walls contribute additively to *k*_{B}, only the outer wall contributes to *k*_{SB} unless the MWNTs are extremely long. Likewise, decreasing the separation between effectively welded inter-nanotube contacts (such as by increasing sheet density) can decrease Poisson's ratio. However, the effects of these structure changes are not simple, because increasing *k*_{B} and decreasing *L* can decrease nanotube meandering between junctions, and this decrease of meandering can provide a positive contribution to *k*_{SB}.

Negative Poisson's ratios are sometimes accompanied by much rarer mechanical properties: negative linear compressibilities and negative area compressibility—meaning that a material expands in either one or two orthogonal directions when hydrostatic pressure is applied (*1*). A negative linear compressibility is the inverse of another strange property, increasing density when elongated in a direction where linear compressibility is negative, and both require that 1 – ν_{1} – ν_{3} < 0. Using the above equations for ν_{1} and ν_{3} as a function of *R* and γ for the model of Fig. 1C, negative in-plane compressibility (Fig. 1D), negative area compressibility for the sheet plane, and stretch densification are predicted for cos γ > (2/3)^{1/2}, which implies γ < 35.3°. However, the average γ needed for achieving these properties will decrease as a result of in-plane nanofiber meandering, because only the tensile strain component resulting in thickness change affects ν_{3}.

The observed continuous tunability of Poisson's ratio, modulus, strength, toughness, density, and electrical conductivity of nanotube sheets could be useful for applications, as could mechanical property optimization using mixtures of nanotubes. However, the change of Poisson's ratio from positive to negative is especially interesting and unexpected. This tunability, which we can now explain, could be exploited in the design of sheet-derived composites, artificial muscles, gaskets, stress/strain sensors, and chemical sensors where analyte absorption induces mechanical stresses. Even shaping processes are affected, because bending a thick nanotube sheet strip will result in either convex or concave lateral deformation (Fig. 1A, inset), depending on the sign of the in-plane Poisson's ratio.

**Supporting Online Material**

www.sciencemag.org/cgi/content/full/320/5875/504/DC1

Materials and Methods

SOM Text

Figs. S1 and S2

References