James W.
Andrews
*a,
James
Bowen
a and
David
Cheneler
b
aSchool of Chemical Engineering, The University of Birmingham, Edgbaston, B15 2TT, UK. E-mail: ANDREWJW@BHAM.AC.UK; Fax: +44 (0)121 414 5354; Tel: +44 (0)121 414 5080
bSchool of Mechanical Engineering, The University of Birmingham, Edgbaston, B15 2TT, UK
First published on 3rd May 2013
An analysis of a novel indentation model has been implemented to obtain master curves describing the optimal experimental parameters necessary to achieve the highest possible accuracy in the determination of viscoelastic properties of soft materials. The indentation model is a rigid indenter driven by a compliant measurement system, such as an atomic force microscope or optical tweezers, into a viscoelastic half space. The viscoelastic material is described as a multiple relaxation Prony series. The results have been extended via an application of a viscoelastic equivalence principle to other physical models such as poroelasticity. Optimisation of the indentation parameters has been conducted over many orders of magnitude of the velocity, viscoelastic moduli, spring stiffness, relaxation times and the duration of indentation resulting in a characteristic master curve. It is shown that using sub-optimal conditions gives the appearance of a more elastic material than is actually the case. For a two term Prony series the ideal ramp duration was found to be approximately one eighth of the relaxation. Also the ideal ramp duration for a three term Prony series was determined and shown to guarantee distinct relaxation times under specific conditions.
An understanding of the mechanical properties from a micro- to macroscopic length scale12,13 is of interest to understand the microstructure of hydrogels. For cases such as these, compared to other techniques like tensile14 and bending tests,15 indentation can be advantageous as it allows for the consistent application of loads,16 the simple acquisition of transient data17 and the sample can be tested in the most convenient environment, be it aqueous or otherwise.18 However the usefulness of this technique at small length scales is limited by the compliance of the measurement equipment,19 which is inherent in many highly accurate systems such as atomic force microscopy (AFM) and optical tweezers (OT). Further the relaxation to the equilibrium state proceeds more rapidly at small length scales.20
Typically, in order to measure transient data, two techniques are employed: creep testing and stress relaxation.21 In these cases, the force applied or indentation depth of the probe is increased to a set value and then held constant whilst the other parameter, depth or force respectively, is monitored. Classically, these techniques in general demand that the maximum force or indentation depth is attained instantaneously in order to allow for simple mathematical analysis using step functions.22 This condition is an idealised situation unattainable experimentally, due to finite acceleration, resulting in experiments where the analysis can only ever be approximate as relaxations in the material that occur during the instrument ramp in force or displacement are neglected.23,24 The appropriateness of the idealised models is diminished further when the compliance of the indentation device is considered. All instruments incorporate an element of compliance, which is usually an inherent property of the measurement system. When indentation is performed using compliant systems, the fixed end of the compliant component is controlled. If possible, this ‘fixed end’ can be driven so that the deflection of the beam is maintained, via feedback control, such that the force applied to the sample increases linearly to a set value and then held so that it resembles a classic creep test.25 However, force control can be difficult to achieve for highly compliant systems. In contrast, a prescribed fixed end motion is experimentally achievable but precludes simple analysis.
The analysis is based on the theory derived for the case of the indentation of a rigid sphere into a planar elastic half space, considered first by Hertz26 and subsequently the indentation of a rigid sphere into a planar viscoelastic half space as considered by Lee and Radok.27 While Lee and Radok27 considered the response of a Maxwell fluid, Yang28 extended the theory to a standard linear solid (SLS) indenting under its own weight. It should be noted that the theoretical analysis of Lee and Yang does not describe the response of the material when the sphere is retracted from the viscoelastic half space. Removing the sphere requires additional constraints to ensure that the stress and strain fields are continuous in time; Ting29,30 developed an approach to deal with this case, but this approach will not be developed further here. The next major development in the theory of creep testing considered the effect of increasing the force to a maximum value in a finite time before it is held constant.31 Incorporating the effect of system compliance has been considered in certain situations, especially for the AFM, yet these solutions are few and involve solving the full beam equation for an oscillating cantilever;32,33 they do not apply for a cantilever which is not oscillating.
Here the indentation of a viscoelastic material by a rigid sphere attached to a compliant indentation system is considered. In order to ensure the mathematics describes the most convenient experimental set-up whereby force feedback is not necessary, nor are instantaneously applied loads, the model assumes the sphere is driven into the material via a compliant element. Furthermore, it is assumed that the fixed end is driven with a constant velocity to a specified distance and held constant; an experimental set-up that is easy to create accurately. The equipment that is frequently used to indent materials is investigated along with an example of each of the four major classes of materials of interest in the literature. This explains why the subsequent developments are necessary to improve the accuracy of measurement of viscoelastic properties. It is shown that the indentation depth and the measured force are coupled in a complex manner that can be described exactly by non-linear Volterra equations. These equations include the effects of the finite time of the ramp, the compliance of the system and the effects of the pre-load which is often unavoidable during experimentation. For the case where the compliance tends to infinity, the exact equations are solved asymptotically. The exact equations do not have an analytical solution and need to be solved numerically. The method for doing so is explained in the ESI [Section 2].†
Analyses of the exact equations show that there are values of the experimental parameters that lead to less than ideal indentation conditions that will result in low accuracy measurements. It is shown that there are ideal experimental parameters that ensure maximum possible accuracy. The procedure for deducing the optimal experiment for a range of machines and materials is presented. Also discussed are the implications of the assumptions used in the derivation of the model, including the assumption of affine deformations implicit in assuming the material is a linear viscoelastic material.
Fig. 1 Standard elements present during an indentation measurement. |
(i) the fixed end, which is usually a motorised, piezoelectrically, or magnetically controlled displacement stage;
(ii) the compliant element or spring, which is frequently the load measuring element i.e. the AFM cantilever or a force transducer, of each device;
(iii) the free end, which is a probe of specific geometry, is attached at the end of the compliant spring and is the element which makes contact with the surface being indented;
(iv) the viscoelastic material being measured.
In the example case of an AFM, the compliant spring is the cantilever beam and the free end is either a pyramidal tip with a sphericonical apex of radius approximately 10 nm, or a spherical colloid probe of radius in the range 0.5–10 μm. For a nanoindenter the compliant spring is often a ceramic pendulum or some kind of high stiffness element, and the free end is a pyramidal, spherical, or sphericonical diamond-coated probe with dimensions on the micron scale or larger. The compliant spring of a mechanical tester consists of a load cell, which is typically piezoelectric or capacitive in design, whilst the probe is a spherical or hemispherical probe typically of radius greater than 1 mm. For optical tweezers the compliant spring is the effective stiffness of the beam of light used to trap a nanoparticle.34,35
Fig. 2(a) is a scanning electron microscopy image of a rectangular AFM cantilever with attached colloid probe, with the free end, fixed end, and compliant spring elements labelled. Similarly, Fig. 2(b) is a photograph of a mechanical tester with attached spherical indenter, in which the free end and compliant spring elements are labelled; the spring is attached to the main body of the instrument, to which the load cell is attached.
Fig. 2 Examples of indentation equipment: (a) AFM cantilever with colloid probe; (b) mechanical tester with spherical indenter. |
Calculation of viscoelastic moduli using instruments such as these is normally achieved by fitting a pertinent theory to the measured force and indentation depth data. In the cases where either the force or probe displacement can be controlled directly via feedback control, the classical creep or stress relaxation theories are appropriate. However, it is usually the case that the force is measured directly whilst the displacement is given by the assumed or measured response of the actuator to a specified input. This means the indentation depth is not monitored directly but rather assumed to be equal to the displacement of the fixed end. The inherent compliance of the system results in the indentation depth actually being unknown. This is a source of error in indentation measurements. In a typical mechanical tester for instance, the force transducer, to which the probe is attached, is suspended from the centre of a cross-head bar which is driven via a lead-screw attached to the ends of the bar to which the displacement sensors are attached. The displacement of the probe is therefore assumed to be the same as the cross-head. As the force transducer is compliant, the true displacement of the probe is not recorded.
Method | Force range (N) | Spring constant (N m−1) | Indenter radius (m) | Drive velocity (m s−1) | Indentation time (s) |
---|---|---|---|---|---|
Optical tweezers (OT) | 10−14 to 10−10 | 10−5 to 10−3 | 10−8 to 10−7 | 10−9 to 10−7 | 10−4 to 10−1 |
Atomic force microscopy (AFM) | 10−11 to 10−6 | 10−2 to 102 | 10−9 to 10−5 | 10−9 to 10−4 | 10−3 to 100 |
Nanoindenter (NI) | 10−4 to 100 | 104 to 105 | 10−6 to 10−3 | 10−8 to 10−5 | 10−2 to 102 |
Mechanical tester (MT) | 10−3 to 104 | 105 to 106 | 10−4 to 10−1 | 10−5 to 10−2 | 10−2 to 101 |
Material | Viscoelastic moduli (MPa) | Viscoelastic relaxation (s) |
---|---|---|
Glassy polymer | 103 to 105 | 10−4 to 10−2 |
Elastomer | 100 to 102 | 10−1 to 101 |
Hydrogel | 10−3 to 101 | 100 to 102 |
Cells | 10−3 to 10−1 | 100 to 102 |
• The sphere is rigid in comparison to the viscoelastic material.
• The viscoelastic material presents a flat, planar surface, and can be considered to be an isotropic, semi-infinite half-space.
• The viscoelastic material can be described using a generalised Prony series model.
• The sphere approaches normal to the surface.
• The radius of the sphere is sufficiently large compared to the indentation depth that the probe geometry can be given by the parabolic approximation.
• The contact between the indenter and the viscoelastic material is frictionless.
• There is no adhesion between the indenter and the viscoelastic material.
• The compliant element is a linear spring, described by a single spring constant.
Fig. 3 Geometry under consideration for spherical indentation into a viscoelastic material. |
Therefore the position, x, of the fixed end takes this form:
(1) |
(2) |
(3) |
Fig. 4 Schematic of standard linear solid model. ηn is the viscosity of the damping component (dashpot), G1n and Gn are the elastic moduli of the restorative force components (springs), α and λ are the stress and strain applied to the element respectively. |
The constitutive equation that describes the behaviour of the SLS model is
ηnG1n + G1nGn+1λ = (G1n + Gn+1)σ + ηn, n = 1…N |
(4) |
(5) |
When there is only one relaxation time τ = τ2. For a two term Prony series the elastic Hertzian case is obtained with the following substitution, G1 = E and G2 → ∞, where E is the shear modulus of the elastic medium, further τ = τr = 0 and the creep modulus is given by, J(t) = 1/E, which is a constant. The viscoelastic Maxwell fluid is obtained by the following substitution, G1 = GM and G2 = 0, where GM is the Maxwell viscoelastic modulus, and the creep modulus is given by, J(t) = 1/GM + t/η.
(6) |
(6a) |
(6b) |
The introduction of an initial indentation depth, h(0), results in an initial preload of F(0) = −kh(0) although this force requires a reference force as only relative forces are measured. Although the initial indentation will be considered in the derivation it will be set to 0 for all results obtained. This preload prior to the start of the experiment is generally unavoidable during real experimentation as it is necessary to first ensure contact with the material. The limit of eqn (6) may be obtained as t → ∞ to obtain the final indentation depth, (h∞), by a direct application of L'hopital's rule.47 In this way the following polynomial is obtained:
Φ = Ak/(G1(Vγ)1/2) |
Γ = Φ(1 + μ) |
Θ = Γ(1 + h(0)/Vγ) |
(7) |
Fig. 5 Variations in the maximum indentation depth provided by eqn (7) as G2 is varied for a Prony series with N = 2. Low μ values correspond to the elastic limit and high μ values correspond to the Maxwell limit. The material parameters G1 and τ are given in Table 4 and the system parameters are supplied in Table 5. |
Parameter | Definition |
---|---|
t/τi | |
β i | τ i /γ |
h/(Vγ) | |
k/(VγG1) | |
μ | |
F/((Vγ)2G1) | |
Φ | Ak/(G1(Vγ)1/2) |
E ∞ | (Vγ)1/2/(Ak(1 + μ)) |
(8) |
Fig. 6 Errors of the asymptotic results compared to eqn (6) and (S1.6) in the ESI,†E∞ is varied by varying the values of G1 (100 to 1010 Pa), G2 (100 to 1010 Pa), V (10−5 to 101 m s−1), γ (10−3 to 102 s), R (10−4 to 102 m), k (10−4 to 106 N m−1) and τ (10−3 to 101 s). The dashed () and the dot dashed () lines correspond to an error of 1% and 10% respectively and data below these lines are considered acceptable within that error. |
The dashed () and the dot dashed () lines correspond to an error of 1% and 10% respectively and data below these lines are considered acceptable within that error. Generally the error scales with log10(E∞), and the trend shown in Fig. 6 shows two of the three important features the ideal curves must possess. There is initially a decrease in the error with decreasing E∞ until the asymptotic solution crosses the analytic solution and the error begins to increase again, and finally the error must tend to 0 as E∞ tends to 0. The value of E∞ which corresponds to the () line is 0.0846, and for the () line 0.2808. Below this value the asymptotic method produces an acceptable solution, in this limit the compliance of the spring may be ignored.
As a demonstration of the model developed earlier, using the typical material parameters in Table 4 and the system parameters from Table 5 the following values for E∞ are obtained in each case; cells, 0.1193 (below 10% error), hydrogel, 1.5901 (above 10% error) and elastomers, 6.2114 (above 10% error). Fig. 7 and 8 demonstrate the indentation curves and force curves for the numerical case detailed above with the asymptotic results for the cells included for comparisons, the asymptotic solution for the cells case is also displayed as it is below 10% error for the indentation depth for the final indentation depth (1.30%), the largest error occurs at h(γ) and has a relative error 8.88%.
Material | G 1 (MPa) | G 2 (MPa) | τ (s) |
---|---|---|---|
Glassy polymer | 9 × 103 | 104 | 10−3 |
Elastomer | 7 | 101 | 100 |
Hydrogel | 5 × 10−1 | 100 | 101 |
Cells | 5 × 10−3 | 5 × 10−2 | 101 |
Parameter | Symbol | Value |
---|---|---|
Radius of sphere | R | 8 μm |
Spring/cantilever stiffness | k | 1 N m−1 |
Duration of ramp | γ | 0.1 τ |
Velocity of fixed end during ramp phase | V | 100 nm s−1 |
Time step | 10−4 s |
Fig. 7 Comparison of the indentation depth as a function of time for the various materials, the black solid line () represents the motion of the fixed end; the cells represented by the dashed line () were obtained by the numerical code, the solid grey line () is the asymptotic fit to eqn (S1.6); the grey dotted line (…) is the hydrogel and the grey dot dashed line () is the elastomer obtained numerically. |
Fig. 8 Comparison of the indentation force as a function of time for the various materials; the cells represented by the grey dashed line () were obtained by the numerical code, the solid grey line () is the asymptotic fit to eqn (S1.6); the grey dotted line (…) is the hydrogel and the grey dot dashed line () is the elastomer obtained numerically. |
However the error in the force corresponding to this is 54.7% at h(γ) and 11.9% for the final indentation depth, this demonstrates that even if the indentation depth is well predicted a greater relative error in the force is possible. The relative error in the force () may be determined from the relative error in the indentation depth (),
(9) |
As an example, E∞ was calculated for the typical hydrogel properties given in Table 4 and system parameters found in Table 5, except the spring stiffness, which is varied and with γ = 0.1 s. The results of this calculation are shown in Fig. 9. As can be seen, a peak value of E∞ is observed and this corresponds to the optimal value E*∞. The peak value, E*∞, may be sought for all parameters, however such a simple parameter can never capture the full complexity of the problem, as it is independent of τ and fails to capture the effects of γ on h(γ). See Section 5.1.2 for further discussion.
Fig. 9 Determination of optimality E*∞: for the typical hydrogel properties obtained from Table 4 and the system parameters in Table 5 except γ = 0.1 s and the spring stiffness which is varied. The grey dashed line () represents the optimal E*∞ = 0.916 corresponding to a spring stiffness of 0.549 N m−1 and the maximum relative relaxation depth, h* = 0.0807. |
A further result may be obtained by noting the following condition,
(10) |
(11) |
Eqn (10) describes the optimal parameters for maximum relative relaxation depth. A significant weakness of eqn (11) is that should γ be varied, h(γ) will vary disproportionately and hence E*∞ and h* will also vary.
The axes in Fig. 10 are,
(12) |
(13) |
Fig. 10 Master curve governing the optimal conditions, the axes are provided by eqn (12)–(15) for a single relaxation, although results are equivalent for Prony series with additional relaxation times. |
It has been found that a master curve to indicate optimal parameters is possible and is provided by the following equation,
(14) |
(15) |
This expression gives the experimentalist the optimum experimental parameters for any material in terms of the remaining parameters. Using this will ensure the relaxation is maximised and ensure that the best possible accuracy is achieved during the indentation experiment. ESI† [Section 4] details the construction of the functional forms found in eqn (14) and (15) for a two term Prony series. For a two term Prony series a natural consequence of this analysis has been the identification of the ideal ramp duration, γ* ≈ τ/8.
When there are two relaxation elements in the Prony series the determination of the ramp duration is further complicated. For convenience we shall consider the mean frequency and the frequency difference to present results; f = (1/τ1 + 1/τ2)/2 and Δf = (1/τ1 − 1/τ2)/2, where both are greater than or equal to zero, hence Δf can at most equal f and it is assumed that τ1 ≤ τ2. A plot of the optimal ramp duration as a function of the frequency ratio Δf/f is presented in Fig. 11. Eqn (10) will have one maximum value with respect to the ramp duration if Δf = 0 since this corresponds to a single relaxation time, however as the frequency difference increases there is a region where there may be at most three local maxima until the frequencies are sufficiently separated to represent only two local maxima located at the isolated local maxima, however since this happens when Δf/f = 1 this corresponds to a single relaxation time again. Determination of a Prony series requires each modulus to be positive36 and this imposes restrictions on the frequency difference which may be observed. When the moduli are equal the restriction is Δf/f ≥ 9/11 and 9/11 corresponds to the position of the discontinuity in the optimal ramp duration observed in Fig. 11. During small frequency shifts Δf/f < 9/11 at optimality, the mean of the frequency is observed and the material may be well approximated by a single relaxation time at 1/f. For frequency shifts larger than this there is a sudden shift in the position of the optimum and the optimal ramp duration is moved significantly towards the optimal of the isolated larger relaxation time. Further the material requires two relaxation times to be described well until Δf/f is sufficiently close to 1 to prevent detection of the smaller relaxation time.
Fig. 11 Optimal ramp duration as a function of the normalised frequency difference is represented by the solid markers. The solid line represents the optimal ramp duration for the slow relaxation time, τ1. The dashed line () represents the optimal time for the fast relaxation, τ2. |
When the conditions for the Prony series to provide positive moduli are not satisfied at a separation of τ1 = 10τ2 (γ* = 9/11), the location of the discontinuous jump occurs at the time separation predicted in Gradowczyk36 (1969), Soussou37 (1970), Park38 (1999) and Park39 (1999). The connection between this approach and that of Gradowczyk36 (1969) is expected since the approach details the minimum theoretical differences in the relaxation times for the determination of different relaxation times.
To deduce the effect of the signal to noise ratio, it is necessary to assume a maximum working indentation depth of hmax. In this case, the maximum velocity for the ramp phase is provided by,
(16) |
(17) |
The ideal drive velocity to maximise for the relaxation phase is provided by the same maximum drive velocity as the ramp phase, Vmax. The minimum drive velocity required for the relaxation phase to be observable, Vrelaxmin is obtained in a similar manner and is provided by,
(18) |
To demonstrate the use of eqn (10) and (11) the maximum indentation depth permissible for an indentation method is required, a simple approximation for this is the limit of Hertz's26 parabolic approximation hmax = 0.1R, although the real maximum is likely to be less than this due to physical limitations of the instrument construction. The maximum noise needs to be estimated for each of the indentation methods and will be taken simply as the following values: AFM/OT, Δh = 0.1 nm, NI Δh = 1 nm and for the MT Δh = 2 μm. This information has been combined to generate Fig. 12a and b and highlight the methods and parameters which are required permissible for determination of viscoelastic parameters for the hydrogel discussed above and in greater detailed in the ESI [Section 5].† The white regions in Fig. 12 may be split into two; above the black line the indentation depth is greater than the maximum permitted by the parabolic approximation, while below the black line the signal to noise ratio is too low and hence is not discernible from the noise, ensuring the indentation is considered not measurable. The permissible region highlighted in grey indicates that for the hydrogel considered and the ramp duration, only AFM can produce optimal indentations. Further the ideal parameters (optimal parameters for the optimal indentation) are obtained from the permissible region with the greatest velocity. The axis on the right corresponds to the logarithm of the spring constant calibrated to be read from the optimal indentation conditions line, to obtain the optimal spring constant at any given point the lines of constant k (), are used to project the point onto the optimal indentation conditions line and hence may be read from the axis on the right. Fig. 12a does not have the force limits of the method included; the limitations of the radius, velocity and spring constant are included. Fig. 12b is restricted to the case of indentation by AFM, however the force limits are added and in this case the permissible region is diminished as a result. Further in Fig. 12b the () lines are lines of constant force and correspond to the maximum and minimum permissible forces for AFM. The permissible regions are obtained from the intersection of inequality constraints with the following objective function,
ϕ = Vγh* | (19) |
Fig. 12 The permissible indentation regions for the hydrogel provided in Table 4 and the system parameters in Table 5 except γ = 0.1 s and k* is provided by eqn (11). The white regions are not possible either due to over indentation or due to low signal to noise ratio. The grey represents regions where it is possible to indent and obtain meaningful results. The dark grey regions represent regions where it is not possible to indent as the parameter range of the method is not available. The black line represents the optimal indentation parameters and this corresponds to the maximum permissible indentation. Lines of constant spring constant are indicated by () and lines of constant force by (). (a) The force limits have not been considered in this particular figure. (b) For the AFM case highlighted as permissible, the force limits have been added to indicate the permissible region with the force limits included. AFM: Atomic Force Microscopy, OT: optical tweezers, NI: nanoindentation and MT: mechanical tester. |
The set of parameters which satisfy the constraints are highlighted grey in Fig. 12b for the hydrogel example. The particular set of parameters which maximise the objective function are called the ideal parameters and may not be unique, for the hydrogel case in Fig. 12b the ideal parameters are unique (V = 17 μm s−1, R = 10 μm and k = 8 N m−1). Since the constraints imposed by each method, the limits of the velocity, radius, spring constant, force and the signal to noise ratio are not linear it is not possible to directly apply a linear programming optimisation49 to obtain the ideal parameters but this is simply resolved by taking the logarithm of all constraints and the objective function. Refer to the ESI for details [Sections 5 and 6].†
Fig. 13 Indentation curves for the hydrogel with a ramp duration of 0.1 s and the following ideal parameters V = 17 μm s−1, R = 10 μm and k* = 8 N m−1, where the spring constant is altered as specified for the sub-optimal cases. |
The sphere is rigid requires that a spherical indenter is available which is effectively rigid compared to the viscoelastic material and it must be attachable for indentation to the method of choice. This assumption is easily satisfied, especially for the materials described in Table 2, as there are many diamond, ceramic, metallic probes and indenters available that are order of magnitudes stiffer than the materials to be tested.
It is assumed that the viscoelastic material presents a flat, planar surface; the viscous nature of the material should always allow this to be achieved given a sufficiently long settling time given appropriate sample preparation, however should the material be curved the reduced radius should be used. Ensuring the sphere approaches normal to the surface is achievable by adequate experimental design. If this assumption is not satisfied a result equivalent to that of Mindlin and Deresiewicz50 is required, but for viscoelastic materials this does not currently exist. The material is assumed to be isotropic; this is a more problematic issue which is related to the assumption that the material undergoes affine deformations. An extension of this theory for viscoelastic fluid behaviour was introduced by Johnson and Segalman51 and is required to consider non-affine deformations. This is another consideration that can be negated with good sample preparation. The viscoelastic material is assumed to be a semi-infinite half-space. In practice this is only ever approximately satisfied as the sample will always have a finite thickness. If the thickness is insufficient, the stress field could interact with the supporting boundary affecting the measured modulus of the material. A rule of thumb is that the material should have a greater thickness than 10 times the maximum indentation depth (h∞).
The parabolic approximation is assumed to be valid when the indentation depth is 0.1R and is based on the assumption used in the original works of Hertz.26 However; techniques exist to extend the analysis beyond this limit.50,52
The contact between the indenter and the viscoelastic material is frictionless, and this will cause the indentation to be greater than for the frictional case; at present the equivalent of the Coulomb friction for viscoelastic materials does not exist.
The indenting object be it a spring, a cantilever, etc. may be treated as a linear spring only under certain conditions, usually under the assumption of small deflections, and these depend strongly on the specific geometry of the compliant element. As such this limit is not pursued here but should be considered for the specific case of interest.
The assumption of negligible adhesion considered throughout this work is only valid for certain systems and should be verified for each case. The adhesive force depends on the area of contact for a viscoelastic material;56 techniques exist in the literature for determining such dependence on contact area but are usually implemented to detect a poroelastic response.56 Adhesion differs significantly from a poroelastic response as an adhesive instability is observed as the probe approaches a surface, or ‘jump to’63 occurs. Elimination of adhesion is possible under liquid conditions but can rarely be ignored for dry contact,64 indeed this is crucial for fouling and in particular biofouling.65 For small length scales adhesion however small can represent a significant contribution to the net force on the spherical indenter, liquid contact with appropriately selected or modified probes; despite this the indentation of cells have been shown many times to be treated well by neglecting adhesion.4,40,44,60,66–73 Hydrogels typically being stiffer than cells are even less prone to adhesion effects as the length scale is typically larger than for cells and this has been demonstrated many times.41–43,74,75 The general treatment of adhesion in a linear viscoelastic material was considered for monotonic indentation by Hui et al.76 (1998) and has subsequently been extended to include retraction and further indentations.77 An approximation to adhesion in elastomers has been considered for a three element Prony series78 and this solution allows a three element Prony series to be fitted to the data and the effects of adhesion obtained as well as the non-adhesive three element Prony series, this analysis is not limited to elastomers. Hence the results here can be converted into results for adhesive contact once the appropriate conversion of the nth term Prony series has been established.78
Optimisation of the indentation parameters for this model over many orders of magnitude of the velocity, viscoelastic moduli, spring stiffness, relaxation times and the duration of indentation results in a characteristic master curve which enables rapid determination of optimal measurement conditions. Analysis of simulated indentation measurements shows that using sub-optimal conditions generates data in which the resolution of the relaxation region is significantly compromised. This gives the appearance of a more elastic material than is actually the case.
For a two term Prony series the ideal ramp duration was found to be ≈τ/8. Also the ideal ramp duration for a three term Prony series was determined. It was observed that during small frequency shifts Δf/f < 9/11 at optimality, the mean of the frequency is observed and the material may be well approximated by a single relaxation time at 1/f. When Δf/f = 9/11 this corresponds to the position of the discontinuity in the optimal ramp duration time. The location of this discontinuity occurs at the same frequency shift observed by Gradowczyk and Moavenzadeh36 (1969) to guarantee distinct relaxation times.
Asymptotic results help to elucidate the behaviour of short duration and/or stiff indentation experiments. The results presented here have implications for researchers using measurement techniques such as atomic force microscopy and optical tweezers.
Footnote |
† Electronic supplementary information (ESI) available. See DOI: 10.1039/c3sm50706h |
This journal is © The Royal Society of Chemistry 2013 |