The handheld micropipette is the most ubiquitous instrument for precision handling of microliter-milliliter liquid volumes, which is an essential capability in biology and chemistry laboratories. The range of one pipette is typically adjustable up to 10-fold its minimum volume, requiring the use and maintenance of multiple pipettes for liquid handling across larger ranges. Here we propose a design for a single handheld pipette adjustable from 0.1 *μ*l to 1000 *μ*l (i.e., 10^{4}-fold) which spans the range of an entire suite of current commercial pipettes. This is accomplished by placing an elastic diaphragm between the existing pipette body and tip, thereby de-amplifying its native volume range while maintaining its simple manual operating procedure. For proof-of-concept, we adapted a commercial pipette (100–1000 *μ*l nominal range) with a selection of rubber sheets to function as the diaphragms and confirmed the accuracy and precision of drawn volumes are within international ISO-8655 standards across the entire 10^{4}-fold volume range. The presence of the diaphragms introduces a nonlinear mechanical behavior and a time-dependency due to heat transfer, however, by model and experiment, these are redressed so as to maintain the pipette’s accuracy and precision.

## I. INTRODUCTION

Precision liquid handling is an essential capability in all wet laboratories. Advances in the convenience, accuracy, and versatility of liquid handling enable researchers to design new protocols and improve the rate of experimentation. The handheld micropipette was invented in the late 1950s^{1,2} and, despite the emergence of automated liquid handling systems, manual pipettes remain the most ubiquitous tool for manipulating microliter-milliliter liquid volumes. Modern handheld pipettes use the same basic piston-driven design as the original, and approximately 1.5 × 10^{6} units are sold worldwide per year.^{3}

A typical micropipette^{4} (Fig. 1(a)) is held in one hand and operated by depressing a spring-loaded push-button with the thumb through a distance *x*. This button is fixed to an internal piston with cross-sectional area *A*, which thereby displaces a volume *V _{p}* =

*Ax*. This displaces liquid at the pipette tip either by direct contact with the piston or more commonly through an intermediary cushion of air as shown. In both cases, the volume of liquid drawn into the tip is approximate to

*V*. The piston stroke is typically adjusted by twisting the thumb button which translates an internal piston stop.

_{p}The volume range of one micropipette is typically one order of magnitude, for example, 1–10 *μ*l, 10–100 *μ*l, 20–200 *μ*l, or 100–1000 *μ*l. This is established by the range of comfortable thumb motion which limits the maximum piston stroke to *x*_{max} ≈ 1 cm; the only mechanical difference determining the pipette’s absolute volume range is the piston cross-sectional area *A*. By convention *x* is adjusted in increments of *x*_{max}/100 because typical manufacturing and assembly tolerances limit the precision of *x* to micrometers. As a result, micropipettes are often purchased in sets that collectively span a large range and handling liquid volumes differing by more than one order of magnitude requires the use of multiple pipettes. Micropipettes with motorized actuation of the piston have a slightly larger volume range (≈1.5–2 orders of magnitude) and are programmable.^{5} For some researchers this is favorable; for others these benefits are outweighed by the additional cost, weight, and complexity. Moreover, a “feel” is required for some delicate pipetting tasks—such as drawing supernatant and dispensing gels—for which a purely mechanical pipette is preferable.

Here we propose a design for a “universal” handheld micropipette (Figs. 1(b) and 1(c)) with a volume range of 10^{4} that covers the entire collective range of commercial micropipettes, i.e., 0.1–1000 *μ*l. Our innovation is a simple modification to the existing mechanical pipette design whereby an elastic diaphragm is placed in front of the piston such that the volume displacement of the piston *V _{p}* is de-amplified at the diaphragm

*V*by compression of an intermediary air volume

_{d}*V*. The liquid at the tip thereby undergoes a volumetric displacement approximate to

_{a}*V*. This serves to scale the pipette’s native resolution and volume range as a function of the air’s compressibility

_{d}*β*and diaphragm’s stiffness

*k*without changing the overall pipette design or operating procedure. The 10

^{4}range would be accomplished in a single pipette by using one piston in select operation with multiple diaphragms of different stiffnesses. For instance, in our schematic (Fig. 1(b)), an additional switch—clicked by the user between the numbered positions—would select the diaphragm to be actuated by the piston; each diaphragm would have a different stiffness and therefore a different range of pipetted volumes.

The efficacy of this design rests on the de-amplification ratio *dV _{d}*/

*dV*being constant for each diaphragm; however, practical ranges of actuation require deformations of the diaphragm and compressions of the intermediary air volume that are finite rather than infinitesimal. The relationship between

_{p}*V*and

_{p}*V*therefore includes a nonlinear behavior arising from the stiffening of the diaphragm and air volume during a piston stroke. And, heat transfer originating from the compression of the air volume causes, over its duration, a time-dependent pressure change of the air volume which in turn causes a time-dependent deflection of the diaphragm.

_{d}In what follows, we develop an analytical model of the diaphragm-piston mechanism which shows that careful choice of system parameters sufficiently minimizes the nonlinear behavior. Regarding the heat transfer, the compression of the air volume is made isothermal simply by adding a fibrous material into the intermediary air volume to shorten the air’s thermal diffusion length. We experimentally validate the model and, for proof-of-concept, perform pipetting experiments using a single commercial 100–1000 *μ*l pipette fitted with an adaptor component that houses a diaphragm cut from a synthetic rubber sheet. Three diaphragms were prepared that de-amplify the pipette by ratios *dV _{d}*/

*dV*= 1/10, 1/100, and 1/1000, effectively turning the pipette into one with respective volume ranges 10–100

_{p}*μ*l, 1–10

*μ*l, and 0.1–1

*μ*l; for each, the accuracy and precision of pipetted volumes are within international ISO-8655 standards.

^{6}Thus, a universal pipette incorporating these few diaphragms could draw and dispense liquid volumes from 0.1

*μ*l to 1000

*μ*l with performance equivalent to that of commercial micropipettes drawing the same volumes. We perceive that a universal pipette would add convenience in adhering to contamination protocols for various fluids and in sharing pipettes between laboratory stations by effectively performing the duty of a complete suite of current pipettes.

## II. DIAPHRAGM-PISTON MODEL

We proceed by developing equations for the diaphragm’s stiffness *k* and air volume’s compressibility *β*, which we then substitute into an equation for the diaphragm displacement *V _{d}* that results from a piston stroke

*V*or a pressure applied to the external side of the diaphragm differing from ambient pressure

_{p}*P*

_{0}by an amount

*P*. The unactuated state of the diaphragm-piston mechanism corresponds to

_{e}*V*=

_{d}*V*=

_{p}*P*= 0 and

_{e}*P*=

_{a}*P*

_{0};

*P*being the pressure of the intermediary air volume which, for practical reasons, vents to the ambient until actuation.

_{a}### A. The diaphragm’s stiffness *k*

The diaphragm deflects due to an applied pressure difference 𝒫 : = *P _{a}* − (

*P*

_{0}+

*P*) and its stiffness is defined by

_{e} The diaphragm is fabricated from a rubber sheet of thickness *t*, which is stretched homogeneously in-plane by an amount *λ* and then clamped by a circular constraint of radius *a*. We estimate the diaphragm’s bending stiffness to be negligible, especially given *λ*, and therefore approximate 𝒫 as resisted purely by the diaphragm’s in-plane tensile stresses. During experiment the maximum stretch experienced by the material elements comprising the diaphragm are all <1.3, so the rubber is treated as a neo-Hookean solid with shear modulus *μ*.^{7} Under these assumptions, the diaphragm’s stiffness *k* and deformed profile *ω*(*r*) to leading nonlinear order in 𝒫 are (see supplementary material, Section I)

Note that the diaphragm stiffens upon deflection, i.e., *k*_{2} > 0, which is the typical behavior for plate structures as stretching of the midplane begins to contribute to load transmission. This stiffening term is 2nd order in 𝒫 because the deformed shape of the diaphragm is symmetric about 𝒫 = 0 and hence *k* must be independent of the sign of 𝒫. This sets the order of approximation for the model overall.

### B. The air’s compressibility *β*

The intermediary air volume changes from ambient pressure and initial volume $P0,V0$ to pressure and volume $Pa,Va$ due to an applied pressure difference *P _{a}* −

*P*

_{0}; its compressibility is defined by

The air behaves as an ideal gas, satisfying

where *γ* reflects the dissipation of heat that is created in the air during compression and is bounded by isothermal (*γ* = 1) and adiabatic ($\gamma =cpcv\u22121$) limits; $cpcv\u22121$ is the ratio of the air’s specific heat capacity at constant pressure to that at constant volume ($cpcv\u22121\u22481.4$ for air at STP).^{8} During compression *γ* reaches some value $\gamma \u2264cpcv\u22121$ and then returns to *γ* = 1 as heat dissipates from the air into its enclosure at a diffusion limited rate over a timescale ∼*L*^{2}/*α*; *α* being the air’s thermal diffusivity and *L* its diffusion length (see supplementary material, Section IV). Clearly *P _{a}* is dependent on

*γ*, hence for the diaphragm-piston mechanism

*V*is dependent on

_{d}*γ*. Therefore

*V*is precise only on timescales longer than that for thermal equilibration; for

_{d}*γ*: 1.4 → 1,

*V*can vary over 20% (see supplementary material, Section IV). Practical air enclosure dimensions for the mechanism are on the order of centimeters (i.e.,

_{d}*L*∼ cm) for which the thermal equilibration is on the order of seconds, which is also the timescale of pipetting a liquid volume.

Reducing the time for thermal equilibration below the timescale of pipetting a liquid volume evidently requires minimizing *L*, which we do by simply adding into the chamber a fibrous material, specifically, steel wool. A volume fraction ≈0.03 of steel wool is sufficient for its temperature change to be negligible upon absorbing the heat from the air for compression ratios up to 0.1, which is the maximum in our experiments (see supplementary material, Section IV). This makes the compression of the air isothermal on the timescale of pipetting and so its compressibility to 2nd order in *P _{a}* −

*P*

_{0}is

At 1st order the air volume stiffens upon compression, i.e., *β*_{1} < 0, and this is the dominant nonlinear behavior for compression ratios ≤0.1 because $\beta 2Pa\u2212P0/\beta 1\u226a1$.

### C. The diaphragm-piston mechanism

Referring to Fig. 1(c), the volume of the intermediary air *V _{a}* may be expressed as

where *V*_{0} is its initial volume at the mechanism’s unactuated state, i.e., *V _{d}* =

*V*=

_{p}*P*= 0 and

_{e}*P*=

_{a}*P*

_{0}. In differential form,

The differential pressure balance across the diaphragm is conveniently already given by Eq. (1); substituting Eqs. (3) and (7) into it, noting that *d*𝒫 = *dP _{a}* −

*dP*, gives

_{e} The diaphragm’s displacement *V _{d}* at any instant is fully specified by

*V*and

_{p}*P*provided the air compression is isothermal. Therefore

_{e}*V*(

_{d}*V*,

_{p}*P*) and the expansion of this function about the mechanism’s unactuated state

_{e}*V*(0, 0) = 0 is

_{d} The partial derivatives in *c*_{i,j} are available in Eq. (8) and *c*_{0,0} = 0. Calculating *V _{d}*(

*V*, 0) explicitly to 3rd order gives

_{p} where *k _{n}* and

*β*are those defined in Eqs. (2a) and (5), respectively. In Secs. III and IV, this is the expression used for comparing the model and experiment and for determining the mechanism parameters (

_{n}*t*,

*a*,

*μ*,

*λ*,

*V*

_{0}) to demonstrate the universal pipette concept. Approximation to 3rd order is necessary to capture the leading-order nonlinear behavior in both the air volume and diaphragm, which are determined experimentally to be significant.

## III. MODEL VALIDATION

To implement and validate the diaphragm-piston concept, we constructed a benchtop measurement setup (Fig. 2(a)) in which a rubber sheet is constrained to function as the diaphragm inside the adaptor component and is deflected by the *V _{p}* provided by a motorized piston; simultaneously,

*P*is recorded with a pressure sensor and the diaphragm’s deformed shape

_{a}*ω*(

*r*) with a laser profilometer (see supplementary material, Sections II and III). Inside the adaptor component, a cylindrical air chamber with radius

*R*= 12.7 mm and adjustable height $h\u22080,30mm$ allows adjustment of

*V*. This chamber, as well as other regions of

_{a}*V*with sufficiently long thermal diffusion lengths, is filled homogeneously with ≈0.03 volume fraction of coarse-grade steel wool. A removable tab on the adaptor allows the air’s pressure

_{a}*P*to equalize with the ambient prior to experiments.

_{a}Using the measurement setup, we took corresponding measurements of *V _{p}*,

*V*, and

_{d}*P*for a diaphragm with parameters

_{a}*μ*,

*t*,

*a*,

*λ*and air volume initially at

*P*

_{0},

*V*

_{0}. The external side of the diaphragm is exposed to the ambient, i.e.,

*P*= 0, so Eq. (10) describes the experiments. A typical result is shown in Fig. 2; here

_{e}*μ*= 0.63 MPa,

*t*= 0.38 mm,

*a*= 9.38 mm,

*λ*= 1.08,

*P*

_{0}= 100.18, and

*V*

_{0}= 22.60 ml. In preparing the experiment, all parameters are known except for the shear modulus

*μ*, which can vary greatly for a synthetic elastomer depending on the process parameters involved in making it. For this reason, all diaphragms were cut from the same roll of black neoprene, thereby fixing

*μ*and

*t*in all experiments. We find

*μ*= 0.63 MPa consistently fits Eqs. (2a) and (2b) to the experiment results within measurement error for all diaphragms we prepared, which collectively comprise several different combinations of

*a*and

*λ*; the fits for Eqs. (5) and (10) are also within measurement error for all experiments (see supplementary material, Section V). It is on this basis that we conclude the validity of the developed model.

Notice in Fig. 2 that the stiffening of both the diaphragm and air volume is significant and compete with one another by, respectively, acting to decrease and increase the slope *dV _{d}*/

*dV*; this can be viewed in the model by artificially setting

_{p}*k*

_{2}= 0 (dotted-dashed lines in Figs. 2(c) and 2(e)) and

*β*

_{1}=

*β*

_{2}= 0 (dashed lines in Figs. 2(d) and 2(e)). For the largest

*V*in Fig. 2(e), the difference between the dotted-dashed and dashed lines is approximately 15% of the experimental value, indicating the importance of accounting for this nonlinear behavior. In Eq. (10), the effect of the air’s stiffening is primarily 2nd order and sets

_{d}*c*

_{2,0}> 0, while the effect of the diaphragm’s stiffening is 3rd order and sets

*c*

_{3,0}< 0 for practical diaphragm parameters.

## IV. PIPETTING EXPERIMENTS

The adaptor component is removable from the measurement setup and fits onto a commercial pipette (VWR VE1000, 100–1000 *μ*l nominal range) (Fig. 3(a)). It interfaces in a manner identical to attaching a pipette tip, which completes the air volume *V _{a}* and

*V*is provided by the piston inside the pipette. Commercial pipette tips attach in the usual way to the other side of the adaptor (see supplementary material, Section V).

_{p}We validated the universal pipette concept by preparing three adaptors (Table I) that de-amplify the pipette’s native volume range by factors 1/10, 1/100, and 1/1000, corresponding, respectively, to volume ranges 10–100 *μ*l, 1–10 *μ*l, and 0.1–1 *μ*l. For each, we drew deionized water from a beaker and measured its volume *V _{l}* with respect to the volume setting indicated on the pipette

*V*(Fig. 3(b)) in a humidity controlled room using gravimetric and optical methods (see supplementary material, Section VI); the error in

_{s}*V*is the difference from the ideal de-amplification (Fig. 3(c)). The accuracy and precision of all pipetted volumes—defined, respectively, as the ±deviation of the mean and standard deviation—are within international IS0-8655 pipetting standards

_{l}^{6}and a detailed listing is in Table S2. The parameter selection for the adaptors, the calibration procedure, and the pipetting precision are discussed below.

The procedure for drawing the liquid volumes *V _{l}* follows the sequential steps in Fig. 1(c). The pipette is first held in hand, unactuated. For the diaphragm, this corresponds to the unactuated state

*V*=

_{d}*V*=

_{p}*P*= 0 and

_{e}*P*=

_{a}*P*

_{0}. Pressing the thumb button on the pipette enacts a piston stroke

*V*that displaces the diaphragm by

_{p}*V*(

_{d}*V*, 0). Inserting the pipette tip into the liquid reservoir entraps within it a cushion of air having volume

_{p}*V*. Releasing the piston back to its initial position

_{c}*V*= 0 retracts the diaphragm and draws a liquid volume

_{p}*V*into the tip. The combined head and capillary pressure from the drawn liquid column constitutes a

_{l}*P*≠ 0 which stretches

_{e}*V*and causes a residual diaphragm displacement

_{c}*V*(0,

_{d}*P*). Removing the tip from the liquid reservoir further modifies

_{e}*V*,

_{l}*V*and

_{c}*P*.

_{e}The resulting volume of liquid inside the pipette tip is

where Δ*V _{c}* is the net volume change of the air cushion inside the pipette tip, i.e., from the moment of entrapment to after the tip has been removed from the liquid reservoir. The complication in this expression is that both

*V*(0,

_{d}*P*) and Δ

_{e}*V*depend on the final

_{c}*P*, which comes about in a complicated way based on the dynamics and wetting of the liquid inside the pipette tip and at the pipette tip’s orifice. However, these contributions are small compared to

_{e}*V*(

_{d}*V*, 0), so from a practical point of view we found it sufficient to simply consider the approximation

_{p}*V*≈

_{l}*V*(

_{d}*V*, 0) for determining the adaptor’s parameters in Table I. Here, Eq. (10) would be the expression for

_{p}*V*(

_{d}*V*, 0) if the piston inside a commercial micropipette operated by having its initial position—and thereby the initial air volume

_{p}*V*

_{0}—the same for all strokes

*V*. However, it is the end stroke position that is the same, so

_{p}*V*

_{0}changes with the piston stroke as $V0=V\u03030+Vp$, $V\u03030$ being a fixed volume of air. Substituting this into Eq. (10) gives

which is the required expression. Notably, $c\u03032,0<0$ and $c\u03033,0<0$ for practical dimensions, rather than *c*_{2,0} > 0 and *c*_{3,0} < 0 in Eq. (10); this is because, by setting a larger *V _{p}* on the commercial pipette, the effect of the air’s stiffening is outweighed by the air becoming more compressible due to its larger volume from repositioning the piston. The parameters in Table I are therefore chosen simply to minimize $c\u03032,0$ and $c\u03033,0$ within the practical constraints imposed by the adaptor’s dimensions. For the chosen parameters, the inaccuracy from this nonlinear behavior is smaller than the imprecision of the pipetted volumes (Fig. 3(c)), so from a practical point of view the de-amplification appears linear (see supplementary material, Section V).

In our pipetting experiments we found that, across the volume range for each adaptor, the systematic inaccuracy of *V _{l}* with respect to the indicated volume setting on the pipette

*V*largely amounts to a translational offset and error in slope; we redressed these, respectively, by adjustment of the calibration mechanism inside the pipette which increments all piston strokes by a fixed amount and by adjustment of the chamber height

_{s}*h*which tunes the de-amplification ratio. This constituted the calibration procedure to produce the pipetting results in Fig. 3.

The primary sources of imprecision in the pipetted volumes *V _{l}* (Fig. 3(c)) are the

*V*(0,

_{d}*P*) and Δ

_{e}*V*that arise due to

_{c}*P*—which for the final static column of water drawn into the pipette tip comprises the addition of head pressure

_{e}*P*and capillary pressure

_{g}*P*

_{σ}, i.e.,

*P*=

_{e}*P*+

_{g}*P*

_{σ}. These scale as

*P*∼

_{g}*ρgH*and

*P*

_{σ}∼

*σ*/

*R*;

*g*is the gravitational constant and

*ρ*,

*σ*,

*H*,

*R*are, respectively, the water’s mass density, surface tension, height, and radius inside the pipette tip. The exact value of

*P*, of course, depends on many confounding dynamic and static factors such as variations in the pipette tip’s geometry and surface energy, contact angle hysteresis, the ambient environment, and the user’s pipetting technique.

_{e}This imprecision can be rationalized based on the following 1st order scaling approximation. If the maximum volume in the adaptor’s range is *V _{L}*, then approximately

*V*∼

_{d}*V*and

_{L}*V*∼

_{c}*ηV*. The factor

_{L}*η*represents approximately how much larger

*V*is compared to

_{c}*V*, and for our adaptor, as well as typically for commercial micropipettes, is

_{L}*η*≈ 2. It follows that $\Delta Vc\u223c\eta VLPeP0$ and $Vd(0,Pe)\u2248c1,0\beta 0Pe\u223cV\u03030VLPeVpP0$, so the total imprecision $Vd(0,Pe)+\Delta Vc\u223cV\u03030Vp+\eta VLPeP0$. Estimates of this quantity are in order-of-magnitude agreement with the results in Figure 3(c).

Most notably, $Vd(0,Pe)\Delta Vc\u223cV\u03030\eta Vp\u22485\u22128$ for our adaptor parameters (Table I), so the majority of the imprecision in the pipetting results is likely due to *V _{d}*(0,

*P*). Therefore designing for larger compression ratios of the intermediary air, i.e., smaller ratios

_{e}*V*

_{0}/

*V*, presumably would improve the pipetting precision. Implementing this would involve making stiffer diaphragms and adding a larger volume fraction of steel wool; the ultimate limit is

_{p}*V*

_{0}/

*V*≈ 1 because the intermediary air volume must at least contain the volume displacement of the piston. Additional potential sources of error not investigated here include the aging and temperature dependence of the diaphragm’s stiffness, as well as ambient pressure fluctuations. We selected neoprene for the diaphragm material because of its durability and availability; however, lifetime requirements and chemical compatibility with common laboratory liquids likely require a different elastomer or different material type such as a corrugated metal foil. Our developed model should apply just as well to any elastomer that can be approximated as neo-Hookean. For other materials, Eqs. (10) and (12) are still correct, however new expressions for

_{p}*k*would have to be derived.

_{n}In regard to volume ranges smaller than 0.1–1.0 *μ*l, which surpass the lower limits of current handheld pipettes, realizing a larger de-amplification ratio using the diaphragm is straightforward. However, the imprecision from capillary pressure *P*_{σ} begins to limit the practicality of the pipette. For instance, the imprecision in the range 0.01–0.1 *μ*l would be at best about ten percent of *V _{L}* according to the above scaling approximation. Other practical limitations also emerge, including the evaporation of the pipetted volumes and additional challenges in performing calibration measurements. All these difficulties apply to current commercial pipettes as well and likely explain why piston-driven pipettes with smaller volume capabilities are not available on the market.

In closing we note that, despite our use of adapters as a proof-of-concept, the universal pipette design could be manufactured at scale without adding significant cost, complexity, weight, or form modification to the handheld micropipette. As depicted in Fig. 1(b), a universal pipette for the 0.1–1000 *μ*l range could be realized with as few as three internal diaphragms (i.e., one per decade below the piston’s volume range). Here, the user would select the diaphragm using the switch on the body of the pipette, and then select the drawn volume proportionally within the diaphragm’s range by adjusting the piston stroke in the usual manner. A universal pipette may provide convenience and efficiency during experiments, simplify maintenance, and reduce inaccuracy due to differences in systematic errors between separately calibrated pipettes. Moreover, the ability to redress systematic errors in both the offset and slope of *V _{l}* with respect to the volume setting

*V*may be a calibration advantage over current pipettes.

_{s}de-amplification . | 1 . | 1/10 . | 1/100 . | 1/1000 . |
---|---|---|---|---|

λ | n/a | 1.241 | 1.212 | 1.250 |

a (mm) | n/a | 9.38 | 4.61 | 2.73 |

t (mm) | n/a | 0.38 | 0.38 | 0.38 |

μ (MPa) | n/a | 0.63 | 0.63 | 0.63 |

$V\u03030$ (ml) | n/a | 15.67 | 10.80 | 12.27 |

P_{0} (kPa) | 101.91 | 102.49 | 101.58 | 101.83 |

T_{0} (C) | 22.2 | 22.1 | 22.0 | 22.5 |

%RH | 40–50 | 40–50 | 40–50 | 40–50 |

de-amplification . | 1 . | 1/10 . | 1/100 . | 1/1000 . |
---|---|---|---|---|

λ | n/a | 1.241 | 1.212 | 1.250 |

a (mm) | n/a | 9.38 | 4.61 | 2.73 |

t (mm) | n/a | 0.38 | 0.38 | 0.38 |

μ (MPa) | n/a | 0.63 | 0.63 | 0.63 |

$V\u03030$ (ml) | n/a | 15.67 | 10.80 | 12.27 |

P_{0} (kPa) | 101.91 | 102.49 | 101.58 | 101.83 |

T_{0} (C) | 22.2 | 22.1 | 22.0 | 22.5 |

%RH | 40–50 | 40–50 | 40–50 | 40–50 |

## SUPPLEMENTARY MATERIAL

See supplementary material for details of mathematical derivations, additional experiments, and experiment methods.

## Acknowledgments

We thank Sheng Jiang for performing preliminary pipetting experiments with an early version of the diaphragm pipette, John Lewandowski for contributing to the CAD model for the rubber sheet stretcher, and Alvin Tan for lending a hand in Fig. S7. J.B. was supported by a Department of Defense National Defense Science and Engineering Graduate Fellowship. Funding for hardware and experiments was provided to A.J.H. by a National Science Foundation CAREER Award (No. CMMI-1346638).