Camille
Duprat
*^{ab},
Hélène
Berthet†
^{b},
Jason S.
Wexler‡
^{b},
Olivia
du Roure
*^{b} and
Anke
Lindner
^{b}
^{a}Laboratoire d'Hydrodynamique (LadHyx), Ecole Polytechnique, 91128 Palaiseau, France. E-mail: camille.duprat@ladhyx.polytecnhique.fr
^{b}Physique et Mécanique des Milieux Hétérogènes (PMMH), UMR 7636 CNRS – ESPCI ParisTech – Université Pierre et Marie Curie Université Paris Diderot 10, rue Vauquelin, 75231 Paris Cedex 05, France. E-mail: olivia.duroure@espci.fr
First published on 24th October 2014
Gels are a functional template for micro-particle fabrication and microbiology experiments. The control and knowledge of their mechanical properties is critical in a number of applications, but no simple in situ method exists to determine these properties. We propose a novel microfluidic based method that directly measures the mechanical properties of the gel upon its fabrication. We measure the deformation of a gel beam under a controlled flow forcing, which gives us a direct access to the Young's modulus of the material itself. We then use this method to determine the mechanical properties of poly(ethylene glycol) diacrylate (PEGDA) under various experimental conditions. The mechanical properties of the gel can be highly tuned, yielding two order of magnitude in the Young's modulus. The method can be easily implemented to allow for an in situ direct measurement and control of Young's moduli under various experimental conditions.
Often, microfluidics provides a versatile and precise technique for fabricating gel particles, capsules or fibers through complex flow patterns^{12,13} or flow lithography,^{14,15} as well as patterned gel substrates.^{16}
The mechanical properties of the gels are paramount to many applications, in particular for cellular biology studies. The elasticity of the gel can be used to initiate differentiated reaction in cells.^{11} In addition, the elastic modulus of the gel can be used to measure the distance between two adjacent crosslinks and hence a measure of the size of the mesh, i.e. the space available between the macromolecular chains, e.g. for drug diffusion.^{8} Knowledge of the evolution of the mechanical properties of the gel during cross-linking also allows for a more precise design of fabrication methods under flow,^{17} and fundamental studies of the transport of flexible objects under flow requires an exact knowledge of their modulus.^{18}
Synthetic hydrogels, such as poly(ethylene glycol) (PEG)-based gels, are among the most commonly used for these applications as their properties (in particular their mechanical properties) can be highly tuned. A common and robust approach to synthesize PEG hydrogels is the photopolymerization of macromolecular PEG chains. The photo-polymerized gel properties can be controlled synthetically (by modifying the monomer structure by adding acrylate or methacrylate groups, or changing the concentration of solvent) and with experimentally tunable parameters such as UV light intensity, exposure time or photoinitiator concentration/structure. Cross-linked gels exhibit entropy-driven elasticity, commonly called rubber elasticity, with a characteristic elastic modulus, which determines their deformation under a given force or flow. The modulus of PEG based gel varies highly with these parameters; consequently, values reported in the literature vary from a couple of kPa to over 10 MPa.^{19–21}
Microfluidic fabrication of gel particles has been widely developed over the past few years, and it is thus important to have a robust method to determine their mechanical properties. Since gel properties are highly dependent on their composition, on the exact fabrication conditions and on their environment, i.e. are extremely set-up dependent, there is a need of a reliable method to assess the mechanical properties of these gels directly in situ, in the conditions of the application. Currently, several methods are employed to characterize gels, but generally require to remove the gel from its environment. Macroscopic tests, such as uniaxial compression or tensile tests, or measurements with a rheometer, require large samples of the gel in a dry environment, which is not always compatible with a microfluidic fabrication. AFM and nanoindentation^{20,21} or rheology are useful and accurate tools, but require external equipment and the extraction of the gel particle outside of the micro-channel. There is thus a need to access the global mechanical properties, such as the Young's modulus or the average mesh size, of the gel, with an in situ microfluidic method.
Microfluidics has been extensively used for rheological measurements,^{22} but much less to provide information on the mechanical properties of soft materials such as gels. We have developed a novel microfluidic based method that directly access the mechanical properties of a gel upon its fabrication. We designed a microfluidic channel containing slots filled with a solution of oligomers and photoinitiator. A free standing beam of precisely controlled rectangular shape is fabricated in situ within these slots (Fig. 1(a–b)), using the stop-flow microscope-based projection photolithography method.^{15,23} It will be used as a probe to measure the mechanical properties of the gel cured in different conditions. When a flow of the uncured solution is imposed in the channel, the beam is pushed into contact with the edges of the slot. The solution then flows over the beam, applying an hydrodynamic force on the gel that deforms accordingly (Fig. 1(c)). Mechanical properties of the gel are deduced from the measurement of the deformation of the beam in response to the flow. This method, where both the load applied to the beam and its geometry are precisely controlled and known, gives a direct access to the Young's modulus of the material. Furthermore, the gel remains in its own solution, thus preventing any swelling or conformation changes and ensuring an accurate determination of its mechanical properties as used in the application.
Fig. 2 (a) Principle of stop-flow photo-polymerization. (b) Schematic of the beam and channel geometries. |
We use an inverted microscope (Zeiss Axio Observer) equipped with a UV light source (Lamp HBO 130W) and an external computer-controlled shutter (Shutter Uniblitz V25). The shutter opening time, t_{uv}, can be precisely controlled. The UV light intensity is kept constant in our experiments. The UV light is filtered through a narrow-UV-excitation filter set (11004v2 Chroma) and is then projected into the channel through a beam-shaped mask. There is a reduction in size between the printed mask and the actual object of width w formed in the channel; the reduction factor corresponds to the magnification of the objective corrected by a factor from the tube lens. With our 10× objective, this reduction factor is about 4. The photo polymerization method requires to focus the projected light into the channel, which is different than visible light. Practically, this means the channel will appear out of focus. The height of the fiber h_{b} is set by the channel height and the height of the inhibition layer. For our channels, this layer is constant and of order 4–6.5 μm. The resulting geometry is described in Fig. 2(b). We explore different geometries listed in Table 1. For a fixed constant channel height, the height of the beam h_{c} is constant and we vary both the width w of the beam and the width L of the channel. We then connect the microfluidic device with rigid tubing to a long syringe (Hamilton 50 μL) mounted on a precision pump (Nemesys, Cetoni) in order to impose well-controlled flow rates Q with a range 0.1 nL s^{−1} < Q <10 nL s^{−1}. Once the flow is established, no significant fluctuations are observed.
h _{c} (μm) | h _{b} (μm) | w (μm) | L (μm) |
---|---|---|---|
42 ± 1 | 34 ± 2 | 29 ± 2 | 300 ± 2 |
68 ± 1 | 55 ± 2 | 22 ± 2 | 210 ± 2 |
265 ± 2 | |||
300 ± 2 | |||
345 ± 2 | |||
27 ± 2 | 345 ± 2 | ||
47 ± 2 | 215 ± 2 | ||
265 ± 2 |
In principle, the method described here could be applied to any gel obtained from a free-radical reaction. Here, we use polyethylene glycol diacrylate (PEGDA) which is a very common hydrogel used in a large variety of applications. We use a solution PEGDA (Aldrich) of average molecular weight 575 with Darocur 1173 (2-hydroxy-2-methylpropriophenone, Sigma) as a photoinitiator at a concentration [PI] varying from 2 to 10 vol%. The viscosity of the uncured solution is η = 47 mPa s, measured with a capillary rheometer at 20 °C.
(1) |
The flow rate per unit length is thus given by
(2) |
We can estimate the pressure gradient across the width of the beam as
(3) |
This pressure gradient generates a force per unit length on the fiber f_{p} = Δph_{b}. The viscous stress on one fiber side is given by
(4) |
f = f_{p} + f_{v} = ληU_{m}, | (5) |
(6) |
The ratio of the pressure to viscous forces is
(7) |
As the confinement increases, i.e. h_{c} − h_{b} decreases, the pressure force becomes predominant. In our case, both forces are of the same order of magnitude, although the pressure force dominates (f_{p}/f_{v} ≃ 8).
We only consider the flow rates where the deflection of the beam varies linearly with the applied force (Fig. 3(d)) and thus the elastomeric beam obeys Hooke's law. Assuming that the load applied on the beam is constant along the length L and equal to f, and considering a slender beam (w ≪ L), we can thus solve the Euler–Bernoulli equation for the beam displacement v(y)
(8) |
v(0) = v(L) = 0 | (9) |
(10) |
(11) |
The maximum deflection is given by
(12) |
Using expression eqn (5) for the load f, we find finally
(13) |
(14) |
The equations can be made non dimensional using Y = y/L and V = v/δ, which gives for the profile
(15) |
Fig. 4 (a) Fiber shapes under different experimental conditions: 5% PI, t_{uv} = 750 ms, L = 220 μm, Q = 50 nl s^{−1} (red); 5% PI, t_{uv} = 750 ms, L = 220 μm, Q = 25 nL s^{−1} (purple); t_{uv} = 200 ms, L = 190 μm, Q = 0.8 nL s^{−1} (yellow/orange); 10% PI, t_{uv} = 350 ms, L = 190 μm, Q = 30 nL s^{−1} (green); 10% PI, 10% PI, t_{uv} = 250 ms, L = 230 μm, Q = 1.7 nL s^{−1} (blue). (b) Rescaled shapes v/δ as a function of y/L. Dashed line is eqn (15). |
Second, we measure the maximum deflection as a function of the flow speed for various geometries under identical experimental conditions, and for a given solution ([PI] = 10 vol%, t_{uv} = 250 ms). We report the renormalized deflection δ/Λ in Fig. 5. All the data collapse onto a single line, in agreement with our prediction eqn (13). The collapse of our data for a large set of geometries confirms the validity of our analytical approach. The only unknown parameter, the Young's modulus E, is obtained by measuring the slope of this line. In the case presented here, we obtain a Young's modulus of E = 690 ± 70 kPa.
Fig. 5 Rescaled deflection δ/Λ as a function of the flow speed U_{moy} for 10% PI, t_{uv} = 250 ms, and all fiber and channel geometries listed in Table 1. The black line is a fit of the experimental data giving a slope η/E = (6.8 ± 0.7) × 10^{−8} s, hence a Young's modulus E = 690 ± 70 kPa. Inset: dimensional data. |
A region of the parameter space is not accessible in the experiments. Indeed, below a critical exposure time, the reaction does not initiate and no gel can be formed. For slightly longer times, above the gel point, a beam starts to form but the resulting beam does not match the shape of the mask, and rather resemble a thin filament. We denote t_{start} the time at which a solid hydrogel beam of the shape prescribed by the mask is formed in the channel. The time t_{start} slightly decreases with increasing [PI], but looking at the gelation point is beyond the scope of our study. All measurements are taken above t_{start}, which also ensures that the elastomer is more homogeneous, which is crucial since our method allows for the determination of the average properties of the gel and which is confirmed by the shape adopted by the beams (Fig. 4).
For a given PI concentration, the evolution of the modulus with t_{uv} is reported in Fig. 6. We note that the Young's modulus increases with increasing exposure time. The modulus increases rapidly with t_{uv}, then saturates and reaches a constant value E = 12.5 ± 1 MPa. In this limit of large exposure times, the Young's modulus does not depend on the UV light intensity, [PI], or exposure time. We can thus compare this saturation value to measurements obtained using conventional measurement techniques. We perform measurements of the mechanical properties of our gel, cured in a UV chamber, using a micro indentation tester (CSM instruments MHT). We obtain a value of E = 11.7 ± 1.5 MPa, which is in good agreement with the value obtained in the microfluidic channel.
Fig. 6 Evolution of the Young's modulus with exposure time t_{uv} for [PI] = 6%. Inset: corresponding evolution of the deflection δ/Λ with the flow speed U_{moy}. |
Increasing the photoinitiator concentration does not change the qualitative evolution of the modulus with exposure time, nor the saturation value. With higher [PI], the modulus increases faster, i.e. an identical value is reached for shorter exposure times t_{uv}. In our experiments, we keep the photoinitiator concentration [PI] ≥ 2%, such that the saturation value of ≃12 MPa is always reached at large exposure times, i.e. the PI concentration is never the limiting factor. However, below a certain threshold of this concentration, the maximum value of the modulus will depend on the photoinitiator concentration as it affects the degree of cross-linking. Drira and Yadavalli^{21} give AFM measurement of the Young's modulus of dry samples of PEG-DA575. For a concentration [PI] = 1.5%, they report a value of E ≃ 12 MPa, which is in good agreement with our measurements. However, they find a value of E ≃ 4 MPa for lower PI concentrations ([PI] ≤ 1%). We can thus estimate a threshold value of [PI] ≃ 1%; above this threshold, the value of the concentration only affects the dynamics of cross-linking.
We measure a wide range of Young's moduli, from 77 kPa to 14 MPa (Fig. 7). We can obtain over two decades in Young's modulus by simply adjusting the UV exposure time, which is easily and precisely varied with a computer-controlled shutter. This provides a convenient control parameter to obtain a gel of desired modulus. PEGDA is thus a versatile material with highly tunable mechanical properties; a single solution with one polymer chain length leads to several decades in moduli. The strong dependence of the modulus on the exposure time also emphasizes the need for an accurate method to measure the Young's modulus in the precise conditions of the experiment.
Fig. 7 Magnitude of the Young's modulus as a function of exposure time t_{uv} for various photoinitiator concentration [PI] (4, 5, 6 and 10 vol%). The lines are guide to the eyes. |
(16) |
Step | Rate constant | Reaction |
---|---|---|
Photolysis | PI + hν → PI* → Ṙ | |
Initiation | Ṙ + M → RṀ | |
Propagation | k _{p} | RM _{n} + M → RM |
Termination | k _{t} | RM _{n} + RM_{m} → RM_{n}M_{m} |
The maximum value of the modulus is reached when M_{c} = M_{w}, i.e. there is exactly one monomer chain between two cross-links. For the PEG-DA 575 used in our experiments, this maximum Young's modulus is E ≃ 14.2 MPa, in good agreement with the experimental value. The slight overestimation arises from the assumption that the chain length is constant (although there is a distribution of monomer chain lengths in the original solution); furthermore, network imperfections lead to a lower value of the average modulus.
We can rationalize the effect of [PI] by looking at the kinetics of the reaction.^{26} The rate at which the monomer is consumed is given by
(17) |
The propagation, initiation and termination are second order reactions, and the polymerization speed R_{p} is given by
(18) |
R_{i} = 2ϕεI[PI]h | (19) |
(20) |
The initial concentration of monomer at t = 0 is noted [M]_{0}, and eqn (20) can be solved to give
(21) |
(22) |
In order to take into account the effect of the concentration of photoinitiator, we write τ = β[PI]^{−1/2}, with a setup constant β. We thus use t_{uv}[PI]^{1/2} to rescale our experiments (Fig. 8). All the data from different [PI] concentrations reasonably collapse onto a single curve. As the reaction time increases, the number of consumed monomers increases according to eqn (21). This directly corresponds to a decrease in the average molecular mass between cross-links M_{c}, and thus an increase in Young's modulus, with the same characteristic time τ.
Fig. 8 Evolution of the Young's modulus with rescaled exposure time t_{uv} [PI]^{1/2} for various [PI] concentrations (2, 4, 6, 8, 10 vol%). |
Experimentally, we find that, before saturation, our data are well described by an exponential law
(23) |
We can further infer the effect of the light intensity on the Young's modulus using eqn (22). The effect is similar as the effect of the PI concentration, i.e. only affects the dynamics of polymerization. For example, a decrease in intensity by a factor 4 leads to an increase in τ by a factor 2, hence an important decrease in E for the same UV exposure time, which explains the high variations of modulus values in different experiments. Previous experiments, on a similar set-up and with a similar PEG-DA solution, albeit with a different light source, were used to give an estimation of the Young's modulus of 63 ± 22 kPa.^{6} By taking the ratio of the light intensities between the two experiments, and assuming all conditions to be identical, we can predict a modulus of 100 kPa in fair agreement with these previous estimations.
In addition, the average mesh size ξ can be estimated from the Young's modulus with the relation E = 3kT/ξ^{3}, which gives for our highly cross-linked gels typical mesh sizes between 1 and 5 nm.
The design of the channel and the choice of the shape of the beam are based on the order of magnitude of the Young's modulus of the material and the viscosity of the oligomer solution, with a design parameter Λ given by eqn (14) that can be adjusted to adapt to the value of the Young's modulus as described in §2.4. In order to keep a constant inhibition layer, the channel has to be all-PDMS. The inhibition layer is measured by letting the beam fall on its side (Fig. 1(b)).
For a given channel (i.e. a given height h_{c}) and a constant inhibition layer (h_{c} − h_{b})/2, one can first tune the width L of the channel to reach a specific range of moduli, since the parameter Λ rapidly varies with the length of the beam, i.e. the width of the channel (Λ ∝ L^{4}). Then, the width of the beam can be easily and externally adjusted by modifying the mask to probe different moduli (Λ ∝ w^{−2}) as long as the beam remains slender (w ≪ L). Reducing the depth of the channel h_{c} allows to probe smaller moduli. Finally, we can note that the confinement has a strong effect (Λ ∝ (h_{c} − h_{b})^{−3}); the thickness of the inhibition layer could be tuned by changing the thickness of the PDMS channel,^{24} although this is not straightforward. Nevertheless, adjusting both length and width allows for the measurement of Young's moduli over several decades, from kPa to MPa.
The fabrication of the beam is identical as the one used in the application (i.e. same light intensity, filters, objective). A controlled flow is then applied, and the deflection δ is measured for increasing fluid velocity in the linear region, i.e. the deflection remains small (δ ~ w). A linear fit of the data directly gives the Young's modulus according to eqn (13). Adjustments of the exposure time t_{uv}, and respective measurements of the deflection, allows to target a specific modulus as needed in the application. In addition, we provide a characteristic kinetics time scale τ that can be obtained by an exponential fit of the data. This setup constant allows for a prediction of the modulus for a given exposure time, which in turn can be measured accurately with the beam method.
This technique can thus be used to both choose the conditions to obtain a target Young's modulus and accurately measure the gel properties under specific conditions. Finally, we believe our method is a convenient tool to study the reaction kinetics of photo-polymerized gels and the dependance of their mechanical properties on various parameters; in particular, the evolution of the modulus with solvent concentration and/or swelling could be directly investigated with our micro-channel.
Footnotes |
† Current address: TOTAL, Avenue Larribau, 64000 Pau, France. |
‡ Current address: Princeton University, Mechanical and Aerospace Engineering, Princeton NJ 08540, USA. |
§ The intensity is expressed in E/(m^{3} s) where E denotes Einsteins, i.e. mol of light, and the molar extinction coefficient is expressed in m^{3} (mol^{−1} m^{−1}). |
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