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Aug 19, 2014 - Raman Thermometry Based. Thermal Conductivity. Measurement of Bovine. Cortical Bone as a Function of Comp

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Yang Zhang School of Aeronautics and Astronautics, Purdue University, 701 W. Stadium Avenue, ARMS 3300, West Lafayette, IN 47907 e-mail: [email protected]

Ming Gan School of Aeronautics and Astronautics, Purdue University, 701 W. Stadium Avenue, ARMS 3300, West Lafayette, IN 47907 e-mail: [email protected]

Vikas Tomar1 Associate Professor School of Aeronautics and Astronautics, Purdue University, 701 W. Stadium Avenue, ARMS 3205, West Lafayette, IN 47907 e-mail: [email protected]

Raman Thermometry Based Thermal Conductivity Measurement of Bovine Cortical Bone as a Function of Compressive Stress Biological materials such as bone have microstructure that incorporates a presence of a significant number of interfaces in a hierarchical manner that lead to a unique combination of properties such as toughness and hardness. However, studies regarding the influence of structural hierarchy in such materials on their physical properties such as thermal conductivity and its correlation with mechanical stress are limited. Such studies can point out important insights regarding the role of biological structural hierarchy in influencing multiphysical properties of materials. This work presents an analytic-experimental approach to establish stress–thermal conductivity correlation in bovine cortical bone as a function of nanomechanical compressive stress changes using Raman thermometry. Analyzes establish empirical relations between Raman shift and temperature as well as a relation between Raman shift and nanomechanical compressive stress. Analyzes verify earlier reported thermal conductivity results at 0% strain and at room temperature in the case of bovine cortical bone. In addition, measured trends and established thermal conductivity– stress relation indicates that the thermal conductivity values increase up to a threshold compressive stress value. On increasing stress beyond the threshold value, the thermal conductivity decreases as a function of increase in compressive strain. Interface reorganization versus interface related phonon wave blocking are the two competing mechanisms highlighted to affect such trend. [DOI: 10.1115/1.4027989] Keywords: cortical bone, thermal conductivity, Raman spectroscopy, compressive stress

Introduction Biological materials such as bone have microstructure that incorporates the presence of a significant number of interfaces in a complex hierarchical structure from nanoscale (10 nm to 1 lm) to macroscale (0.1 mm–10 mm). Bone failure at the macroscale is sensitive to intraspecimen as well as interspecimen variations in architectural features as well as in the material properties [1–4]. Studies have shown that the staggered hierarchical structure of interfaces in bone leads to its unique combination of toughness and strength properties. However, how such an arrangement of interfaces contributes to the physical properties of bone is not clear. Such an understanding is desired for practical applications that involve bone machining such as high speed drilling [5], laser ablation [6], and the curing of cements used in hip replacement [7]. Such applications require an understanding of heat propagation in bone and fundamental microstructural causes that affect such heat propagation, particularly with focus on limiting the damage caused by heat. Such an understanding needs to incorporate effect of temperature and compressive stress and can also point out important insights regarding the role of biological structural hierarchy in influencing correlation between mechanical and physical properties. With this view, the present work focuses on measuring thermal conductivity in mm scale bovine cortical bone samples as a function of compressive stress and temperature changes using Raman thermometry. Measurements are complemented by an analytical model that uses the Raman shift 1 Corresponding author. Manuscript received April 13, 2014; final manuscript received July 7, 2014; published online August 19, 2014. Assoc. Editor: Hsiao-Ying Shadow Huang.

measurements as a function of stress and temperature as input to the model. Experimental values for the thermal conductivity of cortical bone tissue vary widely in the literatures [8–15]. There are numerous possible reasons for the measured variability, which includes differences in samples, wet or dry conditions of measurement, directions of heat flow, variation in experimental procedure, and equipment [8,9]. Lundskog [9] drilled holes in the bone samples to accommodate thermocouples, which affected heat flow through the samples (including possible microcracks in samples). Kirkland [11] and Vachon et al. [10] used a “thermal comparator” for the measurement of thermal conductivity, a device that compares the cooling rates of two heated copper spheres in air. Biyikli et al. [14] used insulation and measured the heat flow directly. Davidson and James [13] improved the experimental method of Biyikli et al. [14] to measure the thermal conductivity of cortical bone and to determine its variation with heat flow direction. Because the directional differences were small, they concluded that the bovine cortical bone could be treated as thermally isotropic. A summary of the discussed studies is presented in Table 1. In comparison to the methods used thus far, Raman thermometry is nondestructive, noncontact, and especially suitable for the measurements in the case of the samples with low thermal conductivity or in the case of samples with dimension smaller than mm. Furthermore, it has been proved to be an effective and accurate tool in the measurement of temperature distribution [16–18] and thermal conductivity of silicon structures as a function of strain and temperature [19–24], which is an attribute that is absent in most measurements performed thus far. Measurements regarding influence of straining/stress on thermal conductivity of cortical bone can supply new information regarding thermomechanics of bone

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Table 1 Summary of comparable studies for bones References

Species

Zelenov [8]

Human

12.8 9.7 9.9

Longitudinal Radial Circumferential

Lundskog [9] Vachon et al. [10]

Human Bovine

Dry specimens Dry specimens Fresh specimens

Kirkland [11] Moses et al. [12]

Bovine and Caprine Equine

3.56 0.601 2.27 0.888–3.08 0.80 0.70  0.58 0.54 0.53 0.2 0.3 0.38 0.54

Davidson and James [13]

Bovine

Biyikli et al. [14]

Human

Chato [15] Current study

Human Bovine

and other hierarchical materials that emulate bone microstructural hierarchy, e.g., nacre. During Raman thermometry measurements, the focused laser spot on the sample surface creates localized temperature increase, which can be detected by the spectrometers by knowing the temperature dependence of the Raman peak position [25,26]. The micro-Raman method for thermal conductivity measurement was developed for the first time by Perichon et al. [19]. Perichon et al. [19] presented the method to explore the relation between Raman peak position and temperature change. Using the corresponding heat transfer models, the Raman spectrum can be related to the thermal conductivity of the material [27]. Raman thermometry has been used extensively for measuring thermal conductivities of thin films, such as silicon thin films [22,28]. This method can be applied to thin films deposited onto a thick substrate, where the thickness of the film should be at least one magnitude higher than the laser spot size. In this way, the effect of the substrate can be neglected. Huang et al. [29–31] extended this method for measurements at submicrometer length scale while taking into account the thermal contribution of the substrate and the interface between the thin film and the substrate. Earlier research work has investigated mechanical properties of the cortical bone, such as elastic modulus and hardness using nanoindentation system [32–34]. However, the focus of the present work offers a new advancement in terms of the effect of straining/stress and temperature on thermal conductivity. In the present work, first the relationship between the Raman shift and temperature change in all examined samples is investigated. This is followed by the development of a relation between the Raman shift and compressive stress change. Through a heat transfer model and the established correlations of Raman shift with temperature and stress, the Raman spectroscopy method is applied to measure the thermal conductivity and stress correlations in examined samples.

Methods In this section, a brief description of the cortical bone sample preparation and the setup of the experiments are presented. The examined cortical bone samples were subjected to mechanical load using a modified nanoindentation platform. Raman spectroscopy system was integrated to the system with laser spot focused onto the lateral sample surface while the compressive loading was applied uniaxially. The local stress and surface temperature was monitored by the Raman spectroscopy system using Raman shift measurement. Simultaneously, the temperature of the sample was also measured using resistance temperature detector 021003-2 / Vol. 5, MAY 2014

Conductivity (W/m K)

Notes

Dry specimens Fresh specimens Dry specimens Longitudinal Radial Circumferential Dry specimens Fresh specimens Fresh specimens Dry specimens Longitudinal

(RTD). Corresponding stress, strain, and thermal conductivity of the sample were calculated based on these parameters. Bone Samples Preparation. Bovine bone has been examined in previous studies focusing on the thermal effects of drilling [5] and cutting [35] on bone with an assertion that it is an acceptable substitute for human tissue [13]. In this study, a bovine femora was obtained from a local butcher shop. After removing the muscle, periosteum, and bone marrow using surgical instruments, the bovine femora bone was soaked in the de-ionized water and then was soaked in the 6% solution of hydrogen peroxide to sterilize for 24 h. After cleaning, samples were cut into three different dimensions using water-jet based cutting. The steps of cutting are illustrated in Fig. 1(a). The longitudinal direction was chosen to be parallel to the growth direction of the bone, the transverse direction was normal to the bone growth direction, and the radial one was orthogonal to both. Samples size with the dimensions of 3  3  3 mm, 2  2  3 mm, and 1  2  3 mm were labeled as samples 1, 2, and 3, respectively. The final optical images of the cortical bone samples and the samples under scanning electron microscope (SEM) are shown in Figs. 1(b) and 1(c), respectively. Figure 1(c) shows structure of the cortical bone sample under the SEM in the longitudinal direction which contains the typical unit of cortical bone-osteon (Haversian system). According to the research of Davidson and James [13], bovine cortical bone could be treated as thermally isotropic. Therefore, the longitudinal direction was chosen as load application direction without loss of generality. After slicing the samples, each sample was heated until it was completely dry. After that each sample was wrapped in a sealed plastic bag and was refrigerated until the measurements. Setup of Experiments. The setup of experiments mainly includes two parts: the compression part to apply mechanical loading and the optical path part for spectroscopic measurements. The schematic diagram of the experiment is shown in Fig. 2. The mechanical load is applied using a nanoindentation platform. The load that can be applied by this platform ranges from 0.1 mN to 500 mN, with the accuracy of better than 0.1 mN. Since the load applied to the sample is uniaxial compressive load, a pin with flat end replaced the indenter. Load calibration was performed before each experiment. For high temperature test, two RTD sensors were attached to the end of the pin, with one acting as the heater, and the other one acting as the temperature sensor. As the mechanical load is applied to the cortical bone samples along the longitudinal axis, the Raman laser is focused onto the side surface of the sample using a 40 objective lens along the transverse axis. Transactions of the ASME

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Fig. 1 Bone samples preparation. (a) Steps of cutting the bone sample; (b) scaled images of the prepared cortical bone samples; and (c) SEM figure of longitudinal direction of the cortical bone sample.

The back-scattered Raman signal is collected by the same objective and sent to a spectrometer (Acton SP2500, Princeton Instruments, Inc., NJ). The laser used in this research is 514.5 nm Arþ laser (Modu-Laser, Inc., UT). The laser was lead to the sample by single mode fiber, a collimator, and a dichroic mirror and then focused by the 40 objective. Thermal Conductivity Measurement by Raman Shift. As discussed before, the temperature of the sample surface can be detected by Raman spectroscopy based on Raman shift measurements. By measuring the laser energy absorbed by the sample and corresponding temperature increase of the laser spot on the sample, the thermal conductivity of the sample can be derived with a heat transfer model. The laser beam diameter of the Raman spectroscopy device ranges from 1 lm to tens of lm [30]. Thickness of the examined samples ranges from 1 mm to 3 mm, which is much larger than the laser spot size. In the case of the sample thickness is more than one magnitude higher than the laser spot size, the isotherms can be assumed to be hemispheric. In this case, the heat transfer across the sample–substrate interface is negligible [19]. In this case, the relationship between the local temperature rise and the absorbed laser power is [36] Kf ¼

2P pdDT

(1)

Here, Kf is the thermal conductivity of the sample; P is the absorbed laser power; d is the laser spot size on the surface of the sample; and DT is the temperature increase of the laser heated spot. DT is determined by Fig. 2 (a) Overview of the experimental setup and (b) a schematic of the optical path of the experiments

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DT ¼ t  Ts

(2)

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where t is the local temperature of the laser heated spot and Ts is the surrounding temperature. In this case, it is also the substrate temperature. Due to the anharmonic terms in the vibrational potential energy, the Raman shift is affected by temperature [37,38]. The intensity of the laser follows Gaussian distribution in the radial direction, which is described by the equation below [29]:   2P 2r 2 (3) IðrÞ ¼ 2 exp  2 pr0 r0 where r0 ¼ d/2 is the radius of the laser spot and P is the laser power. The general heat transfer equation is given as @ 2 tðr; zÞ 1 @tðr; zÞ @ 2 tðr; zÞ þ þ ¼0 @r 2 r @r @z2

(4)

Sample and substrate setup is shown in Figs. 3(a) and 3(b). The heat transfer relation in the cylindrical coordinate system for the setting shown in Fig. 3(b) corresponding to experimental setting shown in Fig. 3(a) below is given by Eq. (4). Noting that the temperature distribution is not changing in the angular (U) direction (due to sample size being significantly bigger than laser spot size), U does not appear in this equation. In order to achieve homogeneous boundary conditions, the temperature was normalized by using the relation tðr; zÞ ¼ Tðr; zÞ  Ts . When the sample is thick (the thickness is a magnitude higher than the laser spot size), it is considered to be a semi-infinite body. In this case, the boundary conditions are described by equations below:   @tðr; zÞ IðrÞhtðr;zÞ 2P 2r 2 h þ tðr; zÞ; ¼ f ðrÞ ¼  ¼ exp  @z k k kpr02 r02 when z ¼ 0; and

Table 2 Properties of Hankel transform f ðrÞ  2 2 a r exp  2

gðqÞ   q2 exp  2 a2 2a H

 2  d f ðrÞ 1 df ðrÞ ¼ q2 Hðf ðrÞÞ þ dr2 r dr

Taking Hðtðr; zÞÞ ¼ Cðq; zÞ and Hankel transform, Eq. (4) becomes d2 Cðq; zÞ  q2 Cðq; zÞ ¼ 0 dz2

After applying boundary conditions Eqs. (5) and (6) become Eqs. (10) and (11) shown below, respectively  2 2 dCðq; zÞ P r q h ¼ exp  0 þ Cðq; zÞ; when z ¼ 0 (10) dz 2pk k 8 lim Cðq; zÞ ¼ 0

(11)

z!1

The solution of Eq. (9) is in the form of Eq. (12) Cðq; zÞ ¼ AðqÞ exp ðqzÞ þ BðqÞ exp ðqzÞ

(12)

Substituting the boundary conditions to Eq. (12) from Eqs. (10) and (11), the results are AðqÞ ¼ 0 BðqÞ ¼

(5)

tðr; zÞ ! 0; when ðr 2 þ z2 Þ1=2 ! 1

(9)

(13) 

r02 q2

P   exp  h 8 2pk q þ k

 (14)

(6) Thus

In order to solve Eq. (4), Hankel transform of order zero is used. Hankel transform is equivalent to two-dimensional Fourier transform. It is also called Fourier–Bessel transform. The Hankel transform pairs are ð1 f ðrÞJ0 ð2pqrÞrdr and (7) gðqÞ ¼ 2p 0

f ðrÞ ¼ 2p

ð1

gðqÞJ0 ð2pqrÞqdq

(8)

Cðq; zÞ ¼

  P r 2 q2   exp qz  0 h 8 2pk q þ k

In order to calculate the surface temperature (when z ¼ 0), the inverse Hankel transform with z ¼ 0 is taken leading to the solution for temperature distributions as

0

where J0 ð2pqrÞ is a zeroth-order Bessel function of the first kind. Some properties of the Hankel transform are listed in Table 2.

(15)

tðr; 0Þ ¼ 2p

ð1

Cðq; 0ÞJ0 ð2pqrÞqdq

(16)

0

Fig. 3 (a) Sample and substrate and (b) front view of sample and substrate in cylindrical coordinate system (r, z)

021003-4 / Vol. 5, MAY 2014

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where I0 ðrÞ is the zero-order modified Bessel function of the first kind. The average temperature at the laser spot is ð 1 r0 t ¼ tðr; 0Þ2prdr (17) pr 2 0 Substituting Eqs. (15) and (16) in Eq. (17), the average temperature at the laser spot turns to be ð 1 r0 t ¼ 2 tðr; 0Þ2prdr pr0 0  ð r0  ð1 2 ¼ 2 2pqCðq; 0Þ J0 ð2pqrÞrdr dq r0 0 0 ð 2 1 Cðq; 0ÞJ1 ð2pqr0 Þdq ¼ r0 0  2 2 ð P 1 1 r q exp  0 ¼ (18) J1 ð2pqr0 Þdq h pkr0 0 8 qþ k I1 ðrÞ is the first order modified Bessel function of the first kind. Equation (18) (local hotspot temperature) cannot be integrated manually. Therefore, it is solved numerically. After substituting the measurable parameters using Raman spectroscopy, the relationship between temperature increase t and thermal conductivity k can be found. Experimental Procedure. Most Raman spectroscopy research on bone has shown Raman shift to be correlated to chemical composition and microstructural properties. Based on literatures [39,40], the two values of Raman shift reported are 952 cm1 and 980 cm1 . The Raman shift Dx can be expressed in terms of wavenumber as,   1 1 (19) Dx ¼  k0 k1 Here, k0 is the excitation wavelength, and k1 is the Raman spectrum wavelength. The approximate value of Raman spectrum wavelength is calculated to be around 541.0 nm. Therefore, during experiments it is enough to scan the wavelength near 541.0 nm for getting the value of wavelength at Raman peak, and, thereafter, calculate the Raman shift of samples using Eq. (19). Before calculating the thermal conductivity, it is necessary to get the correlation between the sample temperature and the Raman shift of examined cortical bone samples. In this case, the Raman thermometry is meant to detect the temperature of the sample. Therefore, low laser power was applied as not to create

detectable localized temperature increase on the sample surface. Low power for Raman experiment of silicon/silicon dioxide films is about 1.1 mW (Arþ ion laser with a wavelength of 514 nm) in some papers, e.g., Refs. [29–31]. But laser power is not completely stable and low power causes low Raman signal and long exposure time. Furthermore, the reflectivity of bone surface is much lower than silicon, so low laser power as 1.1 mW would not produce detectable Raman signal. As a result, the laser with the power 4.0 mW has been used in the experiments which is about one third of the maximum output power of the laser at the objective end. The temperature range in the experiments varied from room temperature to 60  C. During exploration of the correlation of temperature and Raman shift of cortical bone samples, the loading range in the experiments was kept from 0 to 400 mN with the interval of 50 mN. Under different temperature and applied stress values, experiments were performed to measure Raman shift values of the cortical bone samples. After that thermal conductivity of the cortical bone was calculated. For each set of data-point reported, 5–10 repeated tests were performed.

Results and Analysis In Results and Analysis, a description of the experimental setup validation along with experimental measurement of thermal conductivity is provided. Determination of Laser Spot Size. For a specific objective, when the light is well focused, the spot size, d, is calculated as d ¼ k=ðpNAÞ

(20)

where k is the wavelength of the incident light; is the numerical aperture of the objective. However, in real practice, the laser spot size is affected by focusing distance and the quality of incident beam. Therefore, the laser spot size d needs to be determined for each measurement. One method to do that is to scan across a cleaved edge [41]. In the measurement of laser spot size, the laser was focused onto the sample for achieving maximum intensity of the Raman signal. Then, the sample was moved laterally for the laser to scan across the edge. Differentiation of the Raman intensity profile with respect to moving distance leads to the intensity profile of the laser spot. The Boltzmann equation was used to simulate the integral of the Gauss fitting equation and to fit the Raman signal intensity, Fig. 4(a). Thereafter, the laser intensity profile was fitted by Gaussian function expð2x2 =r02 Þ for calculating the laser spot radius r0 [41]. Through measurement of the intensity of reflective laser while moving the sample across the objective, the laser spot size can be calculated. The laser spot size shown in Fig. 4(b) is

Fig. 4 Determination of laser spot size. (a) Fitting of the laser intensity and (b) differentiation of the laser intensity with respect to position and determination of the laser spot size by Gaussian fitting.

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24.5 lm. For most of our measurements, the laser spot size is found to be in the range of 19.13 lm–24.5 lm. If an objective with higher magnification is used, the laser spot size can be further decreased. Determination of Absorbed Laser Power. According to Eq. (18), in order to calculate the thermal conductivity of cortical bone, the laser power P absorbed by the sample needs to be measured. This is determined by measuring laser power at different points of the optical path as shown in Fig. 5. When the output laser power from the laser generator was stable, the laser power at five different locations marked in Fig. 2(b) was measured by using power meter (S140C þ PM100USB, Thorlabs, Inc., NJ). The total laser power I1 coming out from the fiber end of the laser onto sample was mostly reflected by the dichroic mirror, but a portion (I2 ) of I1 transmitted through the dichroic mirror. I4 represents the laser power delivered to the sample. The reflected laser power was derived as I3 after taking account the transmission ratio of the objective and the dichroic mirror. Finally, the absorbed laser power was represented as a function of I2 , which was to be measured in every experiment. Based on the assumption that the absorb power of each optical device is proportional to the intensity of laser, the correlation of laser power at different locations satisfies the following equations: I1 ¼ I2 þ I3 þ Id

(21)

I3 ¼ I4 þ Io

(22)

I40

(23)

I4 ¼

þ Is

I4 I30 ¼ I40 I3 I2 þ I3 0 I5 ¼ I3 I1

(24) (25)

In these equations, I1 –I5 are the intensity of laser at the five points, respectively. I30 and I40 are the intensity of reflected laser at points 3 and 4 shown in Fig. 2(b). Id , Io , and Is are absorbed power of laser by dichroic mirror, the objective, and the samples, respectively. In this part, only I1 –I5 are measured by the power meter, the other parameters are calculated by the equations above. Determination of Raman Shift. Based on the available work in the literatures [39,40], the approximate value of Raman shift of bone is known (952 cm1 and 980 cm1 ). Based on these values the approximate value of Raman spectrum wavelength could be calculated according to Eq. (19). The approximate value of Raman spectrum wavelength is calculated to be around 541 nm. Therefore, during experiments it is enough to scan the wavelength

Fig. 5 Intensity as a function of position in order to measure laser power

021003-6 / Vol. 5, MAY 2014

near 541 nm for getting the value of wavelength at Raman peak, and, thereafter, calculate the Raman shift. The charge-coupled device (CCD) of the Raman spectroscopy system captures Raman spectrum using discretized pixels. More pixels of the CCD will result in more accurate measurement of the Raman peak. However, even with high resolution CCD cameras, this discretized capturing process introduces measurement error in the Raman peak position. The accuracy of the Raman peak detection can be improved by fitting the Raman shift spectrum. The Raman signal of bone satisfies Gauss distribution [42,43]. Therefore, after Gauss fitting of the Raman signal, the Raman peak wavelength of the cortical bone samples is measured. Thereafter, the experimental value of the Raman shift of bone samples can be calculated according to Eq. (19). As shown in Fig. 6, there is a slight difference between the measured peak position and the fitted peak position. Therefore, the peak position of the fitted Gaussian curve was treated as the real Raman shift peak. As Fig. 6 shows, the Raman peak wavelength of the sample is 541.3 nm and the Raman shift is 960.7 cm1 , which matches the value range in the literature [39,40]. Correlation of Temperature and Raman Shift of Cortical Bone. In order to obtain the correlation between temperature and Raman shift of cortical bone, the Raman shift of the sample at different temperature needs to be measured. In this part, a low power laser beam was used so as not to induce additional temperature rise. The peak position of the Raman spectrum shifted as the sample temperature changes. There is a linear correlation between the Raman shift of the Stokes peak and the sample temperature. The Raman shifts at different temperatures were marked as shown in Fig. 7. The relationship between the temperature and the Raman shift of the sample after a linear fitting is Dx ¼ 960:984  0:00873T

(26)

This relationship is used to calculate laser spot temperature change based on measured Raman shift of the samples. Correlation of Stress and Raman Shift of Cortical Bone. It is known that the Raman shift of the cortical bone sample is not only affected by temperature of the sample but also affected by mechanical stress inside the sample. The Raman stress measurement is based on the principle of inelastic interaction between the incident laser and the vibration of crystal lattice [44]. When temperature-induced Raman shift without mechanical loading is measured, the measured Raman shift is solely from temperature effect, as has been done in the last part. However, when measuring stress-induced Raman shift at a specific temperature, the laser power should be chosen as not to create noticeable temperature

Fig. 6

Gauss fitting curve of Raman shift

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relationships at the room temperature for the two samples are expressed, respectively, as Dx ¼ 959:684  12:95  106 jrj; for sample 1 and 6

(27)

Dx ¼ 959:561  6:498  10 jrj

for sample 2

(28)

Dx ¼ 958:724  2:350  106 jrj

for sample 3

(29)

and

The relationships can be used in conjunction with Eq. (26) to calculate localized laser spot temperature increase in the sample and correspondingly the stress dependent thermal conductivity. The decreasing slope of these three linear equation shows that the Raman shift decrease with the increase of stress is different in different range of stress. Reduction rate decreases with the increase in the applied compressive stress. Fig. 7 Correlation of temperature and Raman shift of cortical bone

increase of the sample surface by laser heating. This is calibrated by setting the sample to a constant temperature and by measuring the temperature of the sample surface at different incident laser power by using Raman thermometry. The cut-off laser power was determined when the temperature measured by Raman thermometry appeared higher than the sample temperature. The Raman shifts of the samples 1, 2, and 3 under different stresses were measured as shown in Figs. 8(a)–8(c), respectively. The relationship can be fitted using a linear relationship. It is very difficult to get the stress-Raman shift relation of bone samples theoretically owing to the significant structural hierarchy and heterogeneity in the material. Therefore, it is assumed that the relation between stress and Raman shift is linear based on the similar relations in the case of silicon and other crystals [45]. The

Natural Heat Convection Coefficient. According to Eq. (18), before calculating the thermal conductivity of cortical bone, the natural heat convection coefficient between the cortical bone sample and the surrounding environment needs to be determined. In the setup of the experiment, the cortical bone sample could be treated as vertical plate and the air flow around the sample could be treated as laminar flow. So the Rayleigh number and the Nusselt number are [46,47] RaL ¼

gbðTs  T1 ÞL3 gbtL3 ¼ a a

(30)

The natural heat convection coefficient can be calculated as 1=4

 L ¼ 0:68 þ Nu

0:670RaL

½1 þ ð0:492= PrÞ9=16 4=9

(31)

Fig. 8 Correlation of stress and Raman shift of cortical bone (a) sample 1—3 3 3 3 3 mm, (b) sample 2—2 3 2 3 3 mm, and (c) sample 3—1 3 2 3 3 mm

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 L kair Nu h ¼ L

(32)

In the above equations, g is acceleration of gravity; b is thermal expansion coefficient of air; L is the height of the sample;  is the kinematic viscosity of air; a is the thermal diffusivity of air; and kair is the thermal conductivity of air. After substituting the parameters of air and the sample, the natural heat convection coefficient h is simplified as a function of temperature increase t and is given as  L kair Nu ¼ 9:01 þ 6:2302t1=4 h ¼ L

(33)

Calculation of Thermal Conductivity. Combining Eqs. (18) and (33), with the assistance of Wolfram Mathematica, the relationship between the laser spot temperature increase t and thermal conductivity was calculated and is shown in Fig. 9. As expected, the higher the change in laser spot size temperature, the lower is the thermal conductivity. This relationship can be directly used in calculating thermal conductivity using the experimentally measured values of temperature increase at the laser spot as well as the laser power. Using a combination of these relations, the laser spot temperature change t can be calculated and used in the heat transfer relations described above. Figure 10 shows thermal conductivity as a function of stress for all samples. As shown in the case of sample 1, a linear fit to data indicates that thermal conductivity of the samples increases as a function of stress increase in that range of stress shown. The linear fit relation can be expressed as k ¼ 0:411 þ 1:49  106 jrj

(34)

In the case of sample sizes 2 and 3, the Raman shift and temperature relationship was not obtained due to limitations on the size of RTD sensors. However, Raman shift and stress relation were obtained as expressed in Eqs. (28) and (29). The thermal conductivity data points for sample sizes 2 and 3 were obtained as shown in Figs. 10(b) and 10(c). Figure 10(b) also plots the linear relation fitting based on Eq. (35) from the stress value of 0–0.1 MPa and based on Eq. (36) from the stress value of 0–0.05 MPa obtained using the sample size 2. The linear fitting relation of Eq. (36) was the same as the linear fit to the data points shown in Fig. 10(a), under the same range of stress k ¼ 0:458 þ 4:86  107 jrj

(35)

k ¼ 0:444 þ 1:47  106 jrj

(36)

Figure 10(c) plots the overall linear relation between thermal conductivity and compressive stress from 0 to 0.2 MPa, which is expressed as k ¼ 0:426 þ 3:91  107 jrj

(37)

At the same time, as the thermal conductivity decreases when the stress increases from 0.1 MPa to 0.2 MPa, bilinear fitting is introduced to fit the relation between thermal conductivity and compressive stress from 0 to 0.2 MPa. In the range of stress of 0–0.1 MPa, which shares the same stress range of sample size 2 (Fig. 10(b)), the linear relation is k ¼ 0:373 þ 1:67  106 jrj

(38)

For the stress range of 0.1–0.2 MPa, the linear relation is expressed as k ¼ 0:652  9:22  107 jrj

(39)

Based on the above relations, a clear stress dependence of thermal conductivity in cortical bone can be established. As discussed, measured trends and established thermal conductivity–stress relation indicate that the thermal conductivity values increases and then decrease as a function of increase in compressive strain. As Figs. 10(a)–10(c) show, the slope of the linear relation of thermal conductivity–stress decreases while the stress range expands, which means the rate of change of thermal conductivity has decreased with the increase of compressive stress.

Discussion Calibration of the Spectrometer. The accuracy of the Raman measurement relies on the accuracy of wavelength measurement. For example, an error of 0.1 nm in the wavelength measurement of 541 nm Raman signal can cause 3.77 cm1 error in the corresponding wavenumber conversion. Therefore, the spectrometer was calibrated using Hg light source before each experiment. In order to achieve better accuracy in the wavelength shift calculation, the laser line was also scanned after each set of measurement. Extraordinary caution was taken when scanning the laser line. For the purpose of protecting the CCD camera, the notch filter was always used and the exposure time was kept short (

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