Microscale Creep and Stress Relaxation Experiments with Individual Collagen Fibrils

Abstract

Nanoscale macromolecular biological structures exhibit time-dependent behavior, yet a quantitative understanding of their time-dependent mechanical behavior remains elusive, largely due to experimental challenges in attaining sufficient spatial and temporal resolution and control of stress or strain in conditions that guarantee their molecular integrity. To address this gap, an experimental methodology was developed to conduct creep and stress relaxation experiments with individual mammalian collagen fibrils. An image-based edge detection method implemented with high magnification optical microscopy and combined with closed-loop proportional–integral–derivative (PID) control was implemented and calibrated to apply constant force or stretch ratio values to individual collagen fibrils via a Microelectromechanical Systems (MEMS) device. This experimental methodology allowed for real-time control of uniaxial tensile stress or strain with 27 nm displacement resolution. The overall experimental system was tuned to apply step inputs with rise times below 0.5 s, less than 2.5% overshoot, and steady-state error less than 0.5%. Three individual collagen fibrils with diameters 101–121 nm were subjected to creep and stress relaxation tests in the range 4–20% engineering strain, under partially hydrated conditions. The collagen fibrils demonstrated non-linear viscoelastic behavior that was described well by the adaptive quasi-linear viscoelastic model. The results of this study demonstrate for the first time that mammalian collagen fibrils, the building blocks of connective tissues, exhibit nonlinear viscoelastic behavior in their partially hydrated state.

Introduction

Mammalian tissues have been shown to exhibit time-dependent behavior. While collagenous tissues have been studied extensively, e.g. [1–4], and shown to demonstrate non-linear viscoelastic behavior [5–9], the time-dependent behavior of a key building block in tissue hierarchy, the collagen fibril, is considerably less understood. Collagen fibrils are macromolecular nanofibers with diameters between a few tens to a few hundreds of nanometers and aspect ratios of 100:1 or higher [10]. They are comprised of tropocollagen molecules composed of three ~300 nm long polypeptide chains [11]. Within a collagen fibril, the tropocollagen molecules are staggered and aligned along the fibril, which leads to a 67-nm periodicity, known as the D-band [12]. It is generally accepted that the non-collagenous viscous matrix and the interaction between microscale collagen fibers are responsible for the time-dependent behavior of tissues, with very few studies supporting significant viscoelastic dissipation taking place within individual collagen fibrils [13].

Although several studies have concentrated on the mechanical behavior of collagen fibrils in the past [14–16], very few have prior works have investigated their viscoelastic response [17–20]. Among them, Yang et al. [19] used an Atomic Force Microscope (AFM) to study the stress relaxation of bovine Achilles tendon fibrils. The stress relaxation data were fitted to a two-term Maxwell model. The extracted stress relaxation time constants were related to deformation mechanisms taking place in the collagen fibril: the faster and slower time constants (1.8 s and 63 s) were associated to sliding of microfibrils and collagen molecules, respectively. Shen et al. [18] employed a Microelectromechanical Systems (MEMS) based approach to evaluate the stress relaxation response of collagen fibrils extracted from sea cucumber dermis. Two relaxation time constants (7 s and 102 s) were obtained from a five-element Maxwell-Weichert model. Potential time-dependent mechanisms have been identified as molecular straightening, sliding, and rearranging of intrafibrillar water molecules. These experimental studies did not independently control the extension or force applied to collagen fibrils, hence the collagen fibril specimens were subjected to complicated loading profiles that hinder the direct interpretation of the extracted viscoelastic parameters. Furthermore, because of their simplicity and the ability to use Boltzmann superposition, linear viscoelastic models [18–20] were employed to extract the time constants, which were not shown to be valid in the large range of applied strains and stresses. Improved time-dependent studies of nanostructures demand high resolution position sensing and real time control feedback. To this effect, Scanning Electron Microscopes (SEM) and Transmission Electron Microscopes (TEM) provide high spatial resolution, but non-conducting soft nanomaterials experience electron beam-induced damage that is manifested as mass loss, swelling, radiolysis, and loss of orientation and crystallinity [21–23]. A recent study [24] demonstrated closed loop real time control of MEMS devices for nanoscale testing by using on-chip capacitive position sensing with subsecond positioning accuracy. The method was demonstrated with metallic nanowires that were tested inside a TEM, but could also be used in ambient conditions once a specimen is mounted inside an SEM.

MEMS devices have been used in the past as an effective tool to test nanostructures inside electron microscopes, or under optical microscopes in the case of organic materials that are sensitive to electron beam radiation. In this regard, optical microscopy becomes the imaging tool of choice for biological nanostructures that require aqueous or humid environments. High magnification optical microscopy combined with Digital Image Correlation (DIC) has been shown to provide ~25 nm displacement resolution in experiments with polymeric nanofibers tested with the aid of MEMS devices [25]. Naraghi et al. [26] and Das and Chasiotis [27] utilized this optical metrology method to obtain the viscoelastic response of polyacrylonitrile (PAN) nanofibers and their time-dependent contact mechanics, respectively. However, the use of manual force feedback has limited the calculation of time constants from creep measurements to a few seconds or longer. Shrestha et al. [28,29] used closed loop feedback to monitor the response of PAN nanofibers that upon loading to a target strain were allowed to relax under fixed cross-head opening. In this experimental implementation, both the strain and the stress in the fiber change with time. This technique was used to obtain the short time relaxation modes through linear superposition, which eliminated the need for closed loop feedback to maintain constant stress or strain for creep or stress relaxation tests, respectively. Shrestha et al. carried out typical creep experiments with real-time force control by using a sub-pixel pattern matching algorithm [30]: a predetermined 2D pattern was iteratively searched in a specific window to register changes in pattern positions while bilinear interpolation was used to obtain subpixel displacements with 4 nm resolution. The feedback algorithm speed was limited by the sub-pixel pattern matching rate to two feedback loops per second [28]. Faster real-time control feedback could be achieved via simpler edge detection algorithms for position sensing while taking advantage of high magnification optical imaging. Compared to DIC [31], edge detection has the advantage of low computational cost, easy implementation in LabVIEW [32,33] and no requirement for a speckle pattern [34,35]. Prior studies have used edge detection algorithms to measure the in-plane displacement of microdevices with sub-pixel resolution. Ya’akobovitz et al. [35] used an interpolation-based edge tracking method to measure nanoscale motion with motion sensitivity of 30.8 nm·s⁻¹. Kokorian et al. [36] fitted splines to the average intensity profile of the silicon beam edges in a MEMS adhesion sensor, with reported displacement resolution of 0.6 Å and robustness to measure spikes in the motion of the MEMS structure. However, the extensive calculations involved in this method prohibit real-time feedback.

In this study, we present an optical microscopy based methodology for creep and stress relaxation studies of single mammalian collagen fibrils with dry diameters of the order of 100 nm, which could be used in high humidity conditions or inside clear liquids. An edge detection method combined with simple closed-loop proportional–integral–derivative (PID) control feedback to apply constant force or extension to individual partially hydrated collagen fibrils is implemented for creep and stress relaxation experiments with step inputs. The validity of linear and non-linear viscoelasticity to describe the experimental results is examined. The adaptive quasilinear linear viscoelastic (QLV) model [37,38] is shown to describe well the experimental creep and stress relaxation behavior of collagen fibrils.

Materials and Experimental Methods

Mechanical Behavior of Collagen Fibrils under Uniaxial Tension

Fibrils reconstituted from Type I collagen from calf skin were synthesized according to Williams et al. [39], as also described in a previous work by this group [13]. The reconstituted collagen remained refrigerated at 4 °C, until individual nanofibrils were isolated under high optical magnification (50× objective) using an ultrafine tungsten probe attached to a micromanipulator. In order to isolate individual collagen fibrils, droplets of the collagen solution were diluted with deionized water to decrease the fibril density. Some fibrils from the same vial (fabrication run) were selected for TEM imaging and other fibrils from the same vial were selected for mechanical testing. The collagen fibrils were imaged by a TEM to verify the characteristic 67-nm period of D-banding, Figure 1(a). All collagen fibrils were tested in ambient conditions at 60% relative humidity (RH). Reconstituted Type I collagen fibrils and fibers have been used in similar studies before [40,41] because of the ease with which they can be isolated and their physical resemblance to natural collagen (for example, rat tail collagen). A collagen fibril was tested in uniaxial tension to verify that the mechanical response of the present batch of fibrils is consistent with previous studies of this group using the same type of collagen [13]. The σ-λ plot and the calculated tangent modulus, shown in Figure 1(b), agreed with the mechanical behavior reported in [13].

Figure 1.

(a) TEM image of a reconstituted collagen fibril showing the typical periodic D-banding. (b) Uniaxial tension response and tangent modulus of a collagen fibril. The tangent modulus was calculated by fitting two distinct polynomial functions to the two deformation regimes (Regime I: 0< λ<1.02 and Regime II: λ ≥1.02) of the σ-λ curve. The discontinuity in the tangent modulus at λ = 1.02 is due to the use of two distinct fitting functions.

A prior study by this group [13] showed that the mechanical response of reconstituted mammalian collagen fibrils during the first loading/unloading cycle is substantially different from subsequent cycles, thus requiring mechanical preconditioning to obtain a consistent mechanical behavior as a function of time. This is in agreement with preconditioning protocols established before for tissues [42–45]. In this study, the collagen fibrils were preconditioned by cyclic loading for 10 cycles to the maximum target stress or strain, similarly to [13], before performing creep or stress relaxation experiments. After preconditioning, each fibril was completely unloaded and held for 30 min before performing a test. All tests were conducted in air, at 20°C and 60% RH, which were the laboratory conditions. RH was measured using a hygrometer that was placed close to the collagen fibrils.

Mechanical Testing of Individual Collagen Fibrils

A MEMS device was used to test individual collagen fibrils. The device design follows prior works by this group to study the mechanics of polymeric nanofibers [25,46,47]. The freestanding MEMS devices were fabricated in Run 116 of the PolyMUMPs process (MEMSCAP, North Carolina). They were less than 1 mm long, consisting of four components: a paddle for adhesive gripping via an external probe attached to a piezoelectric actuator, a folded beam load cell that is in series with the fibril specimen, a moving grip that is part of the loadcell, marked as ②, and a fixed grip, marked as ③ in Figure 2. The moving grip is part of the load cell while the fixed grip, and the raised rectangular pads marked as ④, are fixed to a substrate. The low stiffness folded beam load cell (customizable with nominal spring constant 1.2 N/m, and true spring constant 0.8 N/m as calibrated according to [25]) provided high force resolution while ensuring that the folded beam deflection varied linearly with the applied force in the entire range of applied forces, following the analysis in [48]. A collagen fibril was attached to the tips of the sliding mount and the fixed grip with an epoxy adhesive. The nominal gauge length of the tested collagen fibrils was 30 μm. The initial fibril length, L₀, was measured from high magnification optical images recorded during each experiment as the distance between the tips of the moving and fixed grip when the edge-detection program registered a minimum load (0.15 pixels of loadcell opening, see also Section 2.3).

Figure 2.

Surface micromachined MEMS device used for fibril testing. The opening of the folded beam loadcell that is marked by the dashed rectangle is in series with the fibril. The paddle is attached to an external PZT actuator to induce motion. The relative displacement of components ① and ② provides the total deflection of the loadcell beams. A collagen fibril is fixed at the tips of components ② and ③ with an epoxy adhesive. The components marked as ③ and ④ are fixed to the substrate.

During fibril loading, high magnification optical images of the MEMS device (focusing only on the inset in Figure 2) were processed in real-time via an edge detection algorithm to compute the relative position of the components of the MEMS device marked as ① through ④ in Figure 2. The displacement of each component, u_i was calculated by subtracting its initial position, x_i, from the instantaneous position, X_i as:

$$u_i\text{=}{X}_i\text{−}{x}_i.$$ (1)

The fibril stretch ratio, λ, was calculated by dividing the change in length of the fibril, given by the relative displacement of component ② with respect to component ③, by the initial fibril length L₀, as:

$$\lambda =1+\frac{\left|u_2-u_3\right|}{L_0}.$$ (2)

The tensile force carried by the fibril was calculated by multiplying the loadcell spring constant, k, with the relative rigid body displacement of component ① with respect to component ②:

$$F=k\left|u_2-u_1\right|.$$ (3)

Motion Detection of MEMS Device Components by Edge Detection

Edge detection [49,50] encompasses a class of mathematical methods aimed at identifying points in a digital image at which the pixel intensity has a sharp transition. An edge is defined at a point in an image where a significant local change in gray level values occurs [50]. To determine an edge, a digital image is often first filtered to reduce noise and to facilitate subsequent gradient calculations. The gradient of the digital image at each point is then numerically approximated using an algorithm, called edge detector. There are two major categories of edge detectors: Search-based and zero-crossing. The search-based edge detectors (e.g. Roberts Operator [51], Sobel Operator [52], Prewitt Operator [53], Canny Detector [54]) define an edge at the local directional maxima (or minima) of the gradient magnitude of image gray levels. Zero-crossing methods (such as Laplacian or Gaussian [50]) search for zero crossings of the second-order derivative of the image intensity. Subpixel resolution is achieved by interpolating the calculated gradient magnitude or the second-order derivative values. A comprehensive review of edge detectors and their properties can be found in [55].

In this work, an edge detection method for LabVIEW [56] was adapted for real-time measurement and control of the force and extension applied to collagen fibrils. The algorithm was used to detect the edges of the device components shown in Figure 3 by searching for sharp transitions in pixel gray scale values along a line of interest (LOI) or region of interest (ROI). The algorithm first scans the intensity profile pixel-by-pixel along the entire LOI. An edge is located by detecting the local maxima or minima of the first derivative of the intensity profile [55]. The algorithm uses a 3×3 Sobel operator [52] to calculate the gray level gradients in digital images, because of its smoothing properties, better localization of edges and relatively low computational cost [57]. While the Sobel operator, G, calculates the gray level gradients in both the x- and the y-direction, only G_x was used in this work, since the motion of the MEMS device, Figure 2, was constrained in the x-direction. The edges that were normal to the direction of motion were used to obtain the rigid body motion of the respective device components. First, LOIs were defined to include the edges of components ①, ② and ④¹ and the edge detection algorithm identified the initial vertical edge locations within the LOI, which served as reference to begin tracking the motion of these edges in each image frame during specimen loading. An example of this implementation is shown in Figure 3: Two distinct edge strength peaks, corresponding to the maxima and minima of the intensity gradient, were located as the LOI traversed an edge. To ensure consistency of the detected edge, the search direction of the LOI always pointed to the positive direction of the x-axis, and the edge location was identified on the descending edge with negative edge strength which is the local gradient of the gray level intensity calculated using a window of 1×5 pixels.

Figure 3.

Implementation of edge detection method. An example LOI is defined across an edge of component ① in the direction of motion. The gray level intensity (blue curve) corresponds to the intensity at each pixel along the LOI. Two edges defined as the minimum and maximum of the edge strength profile (orange curve) are found when the LOI traverses a physical edge. For consistency, the edge location is identified as the minimum value of the edge strength. The image contrast was intentionally increased to better show the outline of the device components.

An initially identified edge could be lost between consecutive images. Therefore, multiple LOIs were used simultaneously along a component edge to obtain an average coordinate for the edge location and further increase the robustness of tracking. Specifically, the location of an edge was calculated as the average location of 35 points along an edge. The number of LOIs along an edge was limited by the control loop processing time in order to fully utilize all digital images collected at 30 fps, which was the maximum frame rate of the CCD camera, SONY XCD-SX90, at hand. Higher frame rate imaging along with fewer LOIs at each edge would decrease the response time of the close-loop feedback. The displacement resolution using this edge detection technique was evaluated by comparing the rigid body displacements of the MEMS device (without a test specimen), as imposed by a linear piezoelectric actuator with linear error 0.03% [58] and calculated with the aid of DIC, with those computed by edge detection. A displacement ramp in steps of 10 nm and for a total travel length of 5 μm was applied by the linear actuator, while images of the device surface were captured by an optical microscope and were simultaneously processed by the edge detection program in LabVIEW. Figure 4(a) shows the linear ramp displacement imposed by the piezoelectric actuator, as calculated by edge detection program vs. DIC, demonstrating that, on average, the two methods provide the same displacement results. Prior studies by this group have shown an average displacement resolution of 25 nm obtained through DIC [25]. Furthermore, the residuals, Figure 4(b), calculated as the difference between the edge detection value and the values predicted by linear regression applied to the data in Figure 4(a), clustered in a narrow band with standard deviation of 0.15 pixels, which corresponds to a value of 27 nm for the CCD camera sensor and 40× microscope objective used in the experiments.

Figure 4.

(a) PZT-imposed linear displacement ramp, calculated via edge detection, vs. DIC. (b) Residuals of linear regression, calculated as the difference between the edge detection value and the prediction by linear regression applied to the data in (a), with one standard deviation equal to ±27 nm.

The uncertainty in stress and strain is derived from the resolution of rigid body displacements by edge detection. Given that the displacement uncertainly of ±27 nm (Figure 4), the fibril gauge section of 30 μm, the 0.8 N/m force constant of the load cell (Section 2.2), and the average fibril diameter of 112 nm (Table 1), the uncertainty in strain is equal to (27 nm)/(30 μm)·100 = 0.09% ≈ 0.1%, and the uncertainty in stress is equal to (0.8 N/m)(27 nm)/[π/4·112 nm)²] = 2.2 MPa ≈ 2 MPa.

Table 1.

Stress relaxation and creep time constants derived for the adaptive QLV model.

Fibril		Stress Relaxation Times (s)			Creep Times (s)
#	Diameter (nm)	τ1 (s)	τ2 (s)	τ3 (s)	τ1 (s)	τ2 (s)	τ3 (s)
1	101	3.5	26	243	2.1	19	300
2	115	2.2	13	106	3.2	21	136
3	121	3.2	26	784	2.7	22	181
Mean	112±10	3.0±0.7	21±8	377±358	2.6±0.5	21±1	206±85

Closed-loop Feedback Control

To conduct creep and stress relaxation experiments, real-time feedback control was applied to reach and maintain the applied force or fibril extension step input set point. The need for force or displacement closed-loop feedback control stems from the MEMS loadcell that is in series with the fibril specimen whose stress relaxation results in contraction of the loadcell and further extension of the fibril. To mitigate this issue, a PID controller [59] combined with real-time edge detection measurements was implemented for closed-loop feedback control of the applied force or extension through voltage inputs to the PZT actuator. As shown in the block diagram in Figure 5, the controller response is based on the pixel error, e, defined as the difference between the set point, R, and the real-time position, Y, of one or more of the components ①, ② and ④ of the MEMS device, as described in Figure 3. The error signal, e, is converted into a physical length [μm] by multiplying the PID controller parameters with a pixel-to-μm conversion factor. The digital output signal, u, is converted into an analog signal by the D/A converter, amplified and input to the PZT actuator that translates the MEMS device. A new real time feedback position, Y, is then obtained for each image through the camera/edge detection system. The controller specifications were set as follows: rise time less than 0.5 s (to reach 95% of the set point), maximum overshoot less than 5% (of the set point), and steady-state error smaller than 1% (difference between the set point and the steady-state response).

Figure 5.

PID closed loop control block diagram. R is the set point fibril extension or force (in pixels) and Y is the instantaneous measurement by the edge detection algorithm. The PID controller output u is converted into a voltage signal through the gain, K_v. For the specific piezoelectric actuator, K_v was determined experimentally as 1/3 V/μm, through edge detection and in the absence of a test specimen.

Results and Discussion

Tuning of the PID Controller

In practice, a step input stress or strain is approximated by a short ramp with the consideration that the response of a linear viscoelastic material approaches that of an ideal step input after a time interval that is about ten times the ramp time [60]. In the few prior studies, the time-dependent behavior of collagen fibrils was modeled using linear viscoelasticity with two-time constants of the order of 10 s and 100 s [18,19]. To compute the shortest of these two time constants with adequate accuracy, a rise time of 1 s or better is required. Therefore, a rise time shorter than 0.5 s was set as one of the controller specifications. Shorter rise times are also possible with the present experimental system by reducing the number of LOIs at each edge and/or by acquiring images faster with a faster camera.

Given the block diagram in Figure 5, it becomes apparent that there is no straightforward way of identifying a transfer function for the entire test system (Plant). Therefore, the PID controller was initially tuned by using the heuristic Ziegler-Nichols tuning method [61,62] due to its simplicity and easy implementation [63]. The Ziegler-Nichols tuning method begins with a zero value for the integral, K_i = K_p/T_i, and derivative gains, K_d = K_pT_d, where T_i and T_d are the integral and derivative time constants, respectively, while the proportional gain, K_p, is increased from zero to a maximum value at which unsteady oscillations are observed in the steady-state response. This K_p value is defined as the ultimate gain, K_u, and the period of the steady-state oscillation, T_u, is then used to set the PID gains depending on the desired controller response [61]. The parameters derived by the Ziegler-Nichols tuning method are not the optimal controller parameters. Instead, they represent a good starting point for further fine-tuning of the controller. In the present implementation, the proportional gain K_p was first increased from zero to a maximum value of 4 μm/pixel before unsteady oscillations appeared in the fibril steady-state engineering strain response, Figure 6(a). Then, the time period of the steady-state oscillations, T_u, was calculated from the power spectral density, which is the square of the magnitude of the Fourier Transform of the time signal, Figure 6(b). The frequencies corresponding to the three largest power densities were averaged to obtain an oscillation frequency of 4.22 Hz. The reciprocal of this frequency is T_u = 0.24 s.

Figure 6.

PID controller tuning via the Ziegler-Nichols method. (a) For K_p = 4 μm/pixel the fibril strain begins to show unsteady oscillations in the steady-state response, as shown in the inset. This K_p value is defined as the ultimate gain, K_u. (b) Power spectral density of the fibril strain output for K_p = K_u = 4 μm/pixel. The frequencies corresponding to the three largest power spectral densities were used to obtain an average oscillation frequency of 4.22 Hz.

Next, the Ziegler-Nichols method prescribes the classic PID gains as K_p = 0.6K_u = 2.4 μm/pixel, K_i = 1.2K_u/T_u = 20 μm/s/pixel and K_d = (3/40)K_uT_u = 0.072 μm·s/pixel. Furthermore, T_i=T_u/2 and T_d=T_u/8. Using these initial parameters, the controller performance was evaluated through a stress relaxation test of a fibril with an initial target engineering strain of 8%. The rise time was found to be 0.27 s while the overshoot was 7.3% of the target strain, Figure 7. In order to reduce the larger than desired overshoot, the K_i value was decreased (with the K_p and K_d values held constant) until the overshoot was reduced to less than 5% and the rise time did not exceed 0.5 s. Further reduction of the overshoot value by decreasing the value of K_i also increased the rise time. After tuning, the controller gain values that met the required specifications, i.e. K_p = 2.4 μm/pixel, K_i = 10 μm/s/pixel, and K_d = 0.072 μm·s/pixel, were used to obtain the system response for three engineering strain step inputs: 8%, 10.7%, and 13.3%, as shown in Figure 7. In all three cases the rise time was <0.5 s and the overshoot decreased from 7.3% (Ziegler-Nichols parameter) to less than 3.3%. The controller prescribed by the Ziegler-Nichols method provided a rise time that was shorter than 0.2 s, which is advantageous in terms of resolving short material time constants, however, an overshoot of 7.3% may have a significant influence on the loading history [64]. Decreasing the K_i value from 20 μm/pixel·s to 10 μm/pixel·s reduced the overshoot while keeping the ramp time to less than 0.5 s. Finally, as shown in Figure 7, for the same controller gains, the different set point strain values resulted in different rise times due to the material non-linear mechanical response, as shown in Figure 1(b).

Figure 7.

Normalized step input response under closed loop PID control using the Ziegler-Nichols prescribed parameters followed by fine tuning to reduce overshoot.

Evaluation of Experimental Results

Figure 8 shows examples of creep and stress relaxation tests on a collagen fibril with diameter 121±5 nm, performed using the tuned PID controller. In this example, the controller met the target stress/fibril strain with 0.2–0.4 s rise time, while the overshoot was 1.8% of the set point stress and 2.5% of the set point engineering strain. The initial ramp is shown in detail in the insets of Figures 8(a,b). The steady-state error in engineering stress and strain was 0.11±0.16 MPa and −0.007±0.018%, respectively, also meeting the set specifications. Similar curves were obtained for all three test specimens. The steady-state errors in engineering stress and strain were within the measurement uncertainties of 2 MPa and 0.1%, respectively. As shown in Figures 8(a,b), the steady-state stress and strain errors were within 0.1% and 0.5% of the set point values, respectively, which contributed to low noise in measured creep strain and relaxation stress. Low noise is critical for data analysis, as, according to [65], 4% noise in stress relaxation response could result in 8% error in the extracted time constants.

Figure 8.

(a) Step input strain and corresponding stress relaxation response, and (b) step input stress and corresponding creep response of a collagen fibril with diameter 121 ± 5 nm. The steady-state values of the applied engineering strain and stress are 4 ± 0.01% and 164 ± 0.1 MPa, respectively. For both step inputs, the rise time is smaller than 0.5 s (insets). The trend lines in the insets are provided for visual purposes only.

Compared to prior works on microscale creep and relaxation studies of soft materials with manual feedback control [26] or by using coupled creep and stress relaxation methods [18], this real-time edge detection method with closed loop feedback control imposed stress or strain histories that closely approximated step inputs [66], thus facilitating the extraction of time constants of the order of a few seconds. Thus, this optical microscopy-based experimental methodology can provide accurate for time-dependent studies of collagen fibrils, and therefore, could be extended to in vitro experiments in physiologically relevant aqueous environments since the particular MEMS device can be immersed in a liquid cell or a petri dish.

Creep and Stress Relaxation Response of Collagen Fibrils

Three collagen fibrils were tested at different levels of stress and strain to evaluate their time-dependent response. Figure 9 shows an example of three stress relaxation tests at engineering strains 4±0.1% - 16±0.1%, and three creep tests at applied stress 78±2 – 311±2 MPa using a fibril with 121-nm dry diameter. Note that the standard deviation in the values of strain and stress are based on the uncertainty calculations in Section 2.3. The initial finite ramp of ~0.5 s is due to the closed loop control action working to reach the set point strain or stress, Fig. 9(a) and Fig. 9(b), respectively. In stress relaxation tests, the stress initially decayed rapidly, followed by steady-state stress decay, Fig. 9(a). A similar response was recorded for the creep strain which initially increased rapidly as a function of time during a short transient, followed by a steady-state response, Fig. 9(b). Similar results were obtained for the other two fibrils.

Figure 9.

(a) Applied engineering strain and corresponding stress response at short and long times. (b) Applied engineering stress and corresponding strain response at short and long times. For clarity, one out of every five experimental data points is shown for the initial response, and one out of every fifty experimental data points is shown for the long term response.

Fig. 10 (a, d) show the relaxation modulus, $E(t)=\frac{\sigma (t)}{\epsilon }$, and creep compliance, $C(t)=\frac{\epsilon (t)}{\sigma }$, for different fibril strain, ε and stress, σ, values, respectively. As shown, the relaxation modulus and creep compliance are not independent of the magnitude of ε or σ, hence linear viscoelasticity cannot be used to describe the time-dependent behavior of collagen fibrils in the range of ε or σ values applied in this study. Prior studies with individual collagen fibrils tested in fully hydrated conditions [18,19] applied linear viscoelasticity models to fit the experimental data for strain values in the range 5–7% [19] and 14–30% [18], which are similar to the values used in this study. However, in those prior studies, different model parameters were used to fit each response curve. Similarly, the commonly used quasilinear linear viscoelastic (QLV) model proposed by Fung [67,68] cannot be used to describe the nonlinear creep and stress relaxation behavior shown in Fig. 9 (a, b) due to the requirement for a single reduced creep compliance (or reduced stress relaxation modulus) for all applied σ (or ε) values. As shown in the insets in Fig. 10(a,d), this condition was not universally satisfied by our experimental data. Therefore, the more versatile adaptive QLV model developed by Nekouzadeh et al. [37,38] was employed to describe the nonlinear viscoelastic behavior of the collagen fibrils. In this model, for an ideal step input in ε₀, σ(t) and ε₀ are related through a nonlinear relaxation modulus E(ε₀,t) as σ(ε₀,t)= E(ε₀,t)·ε₀. The equivalent Prony series representation of the relaxation modulus E(ε₀,t) is given by:

$$\sigma \left({\epsilon }_0,t\right)={\epsilon }_0\left[E_0\left({\epsilon }_0\right)+\sum _i{E}_i\left({\epsilon }_0\right)\exp \left(-\frac{t}{{\tau }_i}\right)\right],$$ (4)

where E₀(ε₀) is the time-independent elastic modulus, E_i(ε₀) is the elastic modulus associated with the time-dependent shape functions $\exp \left(-\frac{t}{{\tau }_i}\right)$.

Figure 10.

(a) Relaxation modulus (inset: reduced relaxation modulus, i.e. relaxation modulus normalized by the initial value). (b) Initial stress relaxation. (c) Stress response in the entire relaxation experiment. (d) Creep compliance (inset: reduced creep compliance, i.e. creep compliance normalized by the initial value). (e) Initial strain response. (f) Strain response in the entire creep experiment. The solid lines in (b), (c), (e), and (f) are the adaptive QLV model best fit curves. For clarity, one out of every five experimental data points is shown for the initial response, and one out of every fifty experimental data points is shown for the long term response.

Similarly, in a creep experiment the compliance function, C(σ₀,t), relates ε(t) and σ₀ as:

$$\epsilon \left({\sigma }_0,t\right)={\sigma }_0\left[C_0\left({\sigma }_0\right)+\sum _i{C}_i\left({\sigma }_0\right)\left(1-\exp \left(-\frac{t}{{\tau }_i}\right)\right)\right],$$ (5)

where C₀(σ₀) is the time-independent creep compliance and C_i(σ₀) is the creep compliance associated with the time-dependent shape functions $\exp \left(-\frac{t}{{\tau }_i}\right)$.

In practical creep or stress relaxation experiments, it is typical to approximate a step input of stress or strain by a short stress or strain ramp with duration t₀, followed by constant stress or strain [69]. The simplest approach to extract viscoelastic time parameters is by using only experimental data that were obtained after 10 times the rise time t₀ [60], which is referred to as the ten-times rule [69]. To reduce the substantial amount of critical primary creep data loss by using the ten-times rule, the Zapas-Craft method [70] was applied in this study, which extends the data available for fitting up to 0.5t₀. According to this method, the creep or stress relaxation data after the initial ramp, e.g. Figures 9(c,g), are shifted along the time axis by half the rise time, (−0.5t₀). Then, using nonlinear regression analysis via the nlinmultifit module in MATLAB [71], the parameters E_i(ε₀), E₀(ε₀), C_i(σ₀), C₀(σ₀), are estimated for each applied ε₀ or σ₀ value. Note that the values for τ_i are held the same for all values of ε₀ or σ₀ applied to a fibril. As shown in Fig. 10 (b, c) and Fig. 10 (e, f), the adaptive QLV model could fit well all the stress relaxation and creep strain data obtained from the same fibril.

In the adaptive QLV model, the time constants, τ_i, are assumed to be independent of the applied stress or strain (but different for creep and stress relaxation). Herein, three time-dependent shape functions (i.e. three time constants) were used, which yielded better fitting to the experimental data than using two time constants. The extracted time constants from three collagen fibrils are given in Table 1. The shortest time constant τ₁ (3.0±0.7 s for stress relaxation and 2.6±0.5 s creep) is comparable to the average short time constants (1.8±0.2 s in [19] and 7±2 s in [18]) reported in previous experimental studies, utilizing linear viscoelastic models, of single collagen fibrils. On the other hand, the average long time constants reported before (63±23 s in [19] and 102±5 s in [18]) lie between the two time constants, τ₂ and τ₃, (21±8 s and 378±358 s for stress relaxation and 21±1 s and 206±85 s for creep) calculated in this study. It should be noted that, in prior studies, different time constants were fitted to different creep (or stress relaxation) data obtained at different applied stress (or strain) values, and the average of those time constants was reported [18,19]. Differences in the extracted time constants by studies could also be due to differences in the source of collagen fibrils (sea cucumber dermis in [18] and bovine Achilles tendon in [19]), and the test conditions of collagen fibrils (partly hydrated vs. in-vitro conditions.) RH and hydration play an important role in the structure and mechanical properties of collagen fibrils. At 60% RH as in the present study, the collagen fibrils are considered partially hydrated with roughly 90% occupancy of specific sites by water molecules [72]. The number and strength of hydrogen bonds is also in partially hydrated conditions [73] because of the reduced fibril diameter compared to in vivo (fully hydrated) collagen, which further contributes to the large elastic modulus and ultimate tensile strength of partially hydrated collagen fibrils compared to in vitro studies.

The fitted relaxation modulus and creep compliance parameters are listed in Table 2. As shown earlier in Figure 1, partly hydrated collagen fibrils exhibit a parabolic dependence of the tangent modulus with strain. Motivated by this observation, second-order polynomial functions for E₀(ε₀), E_i(ε₀) and C₀(σ₀), C_i(σ₀) were utilized in Equations (4) and (5), and the constants calculated through non-linear least squares fitting are given in Table 3.

Table 2.

Relaxation modulus and creep compliance fitting parameters of adaptive QLV model.

Fibril 1
Applied Strain (%)	9	12	16	Applied Stress (MPa)	293	325	380
E0 (GPa)	3.13	2.55	2.25	C0 (GPa−1)	0.28	0.31	0.34
E1 (GPa)	0.16	0.12	0.15	C1 (GPa−1)	0.02	0.02	0.03
E2 (GPa)	0.12	0.10	0.06	C2 (GPa−1)	0.02	0.02	0.02
E3 (GPa)	0.13	0.11	0.13	C3 (GPa−1)	0.01	0.01	0.02
Fibril 2
Applied Strain (%)	6	12	20	Applied Stress (MPa)	81	163	244
E0 (GPa)	2.67	2.09	1.83	C0 (GPa−1)	0.39	0.25	0.27
E1 (GPa)	0.59	0.64	0.27	C1 (GPa−1)	0.05	0.07	0.05
E2 (GPa)	1.01	0.54	0.19	C2 (GPa−1)	0.09	0.04	0.05
E3 (GPa)	0.83	0.33	0.15	C3 (GPa−1)	0.09	0.07	0.03
Fibril 3
Applied Strain (%)	4	8	16	Applied Stress (MPa)	78	156	311
E0 (GPa)	3.19	2.07	1.44	C0 (GPa−1)	0.41	0.26	0.30
E1 (GPa)	0.80	0.76	0.74	C1 (GPa−1)	0.05	0.07	0.06
E2 (GPa)	0.53	0.64	0.30	C2 (GPa−1)	0.08	0.06	0.04
E3 (GPa)	0.30	0.96	0.38	C3 (GPa−1)	0.07	0.04	0.03

Table 3.

Parameterized functions of relaxation modulus and creep compliance derived from the adaptive QLV model, using the fitting results in Table 2.

Fibril 1
E0	126ε2 − 43ε + 6	C0	−4.3σ2 − 3.5σ − 0.4
E1	23ε2 − 6ε + 1	C1	2.3σ2 − 1.5σ + 0.2
E2	−5ε2 + 0.4ε	C2	−1.6σ2 + 1.1σ − 0.2
E3	15ε2 − 4ε	C3	1.8σ2 − 1.1σ + 0.2
Fibril 2
E0	47ε2 − 18ε + 4	C0	11.8σ2 − 4.6σ + 0.7
E1	−41ε2 + 8ε	C1	−3σ2 + σ
E2	26ε2 − 13ε + 2	C2	3.5σ2 − 1.4σ + 0.2
E3	45ε2 − 17ε + 2	C3	−1.5σ2 + 0.1σ + 0.1
Fibril 3
E0	175ε2 − 50ε + 5	C0	9.2σ2 − 4.1σ + 0.7
E1	7ε2 − 2ε + 1	C1	−1.4σ2 + 0.6σ
E2	−61ε2 + 10ε	C2	0.7σ2 − 0.5σ + 0.1
E3	−203ε2 + 41ε − 1	C3	1.8σ2 − 0.9σ + 0.1

It is thus shown that mammalian collagen fibrils demonstrate stress and strain-dependent creep and stress relaxation behavior, respectively, which is largely consistent with the time-dependent mechanical behavior of macroscale connective tissues [7–9]. At the length scale of individual collagen fibrils, although prior works have studied aspects of time-dependent behavior such as energy dissipation [13,74], strain-rate sensitivity [17,75], and stress relaxation [17,75], this is the first time that direct creep or stress relaxation experiments were carried out with nanoscale collagen fibrils. Furthermore, compared to prior experimental studies, it is shown that, in the range of the imposed strain values, collagen fibrils demonstrate a non-linear viscoelastic response. Mechanistically speaking, the shorter stress relaxation (3.0±0.7 s and 21±8 s) and creep time constants (2.6±0.5 s and 21±1 s) derived herein could be attributed to unwinding and rearrangement of molecular kinks [76,77] while the longer stress relaxation (378±358 s) and creep time constants (206±85 s) could be related to sliding between collagen molecules and microfibrils [75]. Through molecular dynamics simulations, Gautieri et al. [78] showed that single collagen molecules respond in a non-linear viscoelastic manner with relaxation time of the order of nanoseconds. While the time for rearrangement or straightening of an individual molecular kink is inferred to be of the order of nanoseconds, not all molecular kinks in a single collagen fibril straighten at the same time, and as a result, the stress relaxation in an individual fibril can be viewed as a cascade of molecular relaxation events occurring along the fibril, which result in a broad spectrum of viscoelastic time constants.

Uncertainty in Extracted Time Constants due to Finite Ramp Loading

Although the rise time in the step stress or strain inputs was limited to less than 0.5 s, the loading profile was comprised of an initial finite ramp, which is not taken into account in the derivation of Equations (4) and (5). To assess the uncertainty in the extracted time constants due to the finite ramp-time, the stress relaxation response of a general linear viscoelastic solid subjected to a step input strain, with and without an initial ramp of duration t₀, was compared. To keep the calculations simple, a three-term linear Maxwell model with one time constant, τ, that was of the order of the shortest time constant extracted from the experimental data, was employed. A single time constant model was chosen because the largest uncertainty occurs in the shortest time constant [79]. For a step input strain, ε₀, the stress relaxation is given by:

$$\sigma (t)={\epsilon }_0\left[E_0+E_1\exp \left(-\frac{t}{\tau }\right)\right],$$ (6)

where E₀, and E₁ were given the values of 3 GPa and 0.8 GPa, respectively, similarly to the results obtained in Section 3.3 (Table 2, Fibril #3) and the input step strain, ε₀, was set to 0.1. Using Equation (6), the stress response for an ideal step input with time constants, τ, in the range 0.5–5 s and 100 s was generated in MATLAB. If the same three-element linear Maxwell material is subjected to a ramp strain with duration t₀ followed by constant strain ε₀ (ramp-and-hold input), the stress response is given by [69]:

$$\sigma (t)=\frac{{\epsilon }_0}{t_0}\left\{E_0{t}_0+E_1\tau \exp \left(-\frac{t}{\tau }\right)\left[\exp \left(\frac{t_0}{\tau }\right)-1\right]\right\}.$$ (7)

The stress response due to the ramp-and-hold input is then shifted by −0.5t₀ to apply the Zapas-Craft method [70]. As an example, the stress response due to an ideal step input and τ = 2 s, Equation (6), is compared in Figure 11(a) to the stress response due to a ramp-and-hold input with t₀ = 1 s, Equation (7), and the stress response after the Zapas-Craft shift. The shifted stress response, starting at 0.5t₀, matches well the stress response due to a step strain input, with 0.15% error (as compared to 4.1% error of the original, unshifted, ramp-and-hold response) at t = t₀, as shown in the inset in Figure 11(a).

Figure 11.

(a) Simulated stress relaxation for step vs. ramp-and-hold input strain with ramp time t₀ = 1 s, and shifted response from the ramp-and-hold input using the Zapas-Craft method. τ = 2 s. The inset shows the relative error in the stress response between the step input and (i) the stress relaxation from the ramp-and-hold input (black line) and (ii) the shifted stress relaxation from the ramp-and-hold input using the Zapas-Craft method (red symbols). (b) Error in the fitted time constants using the Zapas-Craft method vs. t₀. Inset: Error in the fitted value of the time constant, τ, using the original (unshifted) ramp-and-hold stress response and τ = 0.5 s.

This simple linear viscoelastic model provides first-order estimates for the uncertainty in the extracted time constants obtained from the original unshifted ramp-and-hold stress response and the same data shifted by the Zapas-Craft method [70]. The time constant, τ, was extracted from the original (unshifted) ramp-and-hold and the shifted stress response, and compared with the prescribed value under different ramp times, t₀. As shown in the inset of Figure 11(b), a ramp-time of 0.5 s, which is consistent with the tuned PID controller, would result in 4.2% error in the time constant value, τ = 0.5 s, by fitting the raw stress response. This uncertainty is reduced to 0.3% for the same ramp-time t₀ = 0.5 s when the shifted stress response according to the Zapas-Craft method is fitted, Figure 11(b). The error by fitting the shifted stress response further decreases for larger time constants and t₀ = 0.5 s. Based on Figure 11(b), the uncertainty in the extracted time constants for creep (2.6±0.5 s) and stress relaxation (3.0±0.7 s) for a 0.5 s ramp time, Section 3.3, is estimated to be less than 0.1%.

Conclusions

An experimental methodology utilizing a custom-designed MEMS device and a real time edge detection algorithm was developed to investigate the time-dependent mechanical behavior of single collagen fibrils through step stress or strain inputs. The calibrated PID controller allowed for real-time engineering strain and force control, ramp times within 0.5 s and steady-state error below 0.5%. Creep and stress relaxation experiments revealed that collagen fibrils exhibit nonlinear viscoelastic behavior that could be simulated by the adaptive QLV model using three time constants. The uncertainty in the shortest extracted time constants due to the finite (<0.5 s) ramp time was estimated to be within 0.1% by using a three-term linear Maxwell model and the Zapas-Craft shift method. The fitted strain-dependent relaxation modulus and stress-dependent creep compliance parameters exhibited a parabolic dependence on the applied strain or stress. The results of this study demonstrated for the first time that, similarly to collagenous tissues, partially hydrated mammalian collagen fibrils exhibit a nonlinear viscoelastic behavior.

Highlights.

• Optics-based methodology for stress relaxation and creep tests of single nanofibers

• Real time control of MEMS devices used for testing individual nanofibers

• Showed that mammalian collagen fibrils exhibit nonlinear viscoelastic behavior

• Identified suitable non-linear viscoelastic models for individual collagen fibrils