0
Research Papers

Raman Thermometry Based Thermal Conductivity Measurement of Bovine Cortical Bone as a Function of Compressive StressOPEN ACCESS

[+] Author and Article Information
Yang Zhang

School of Aeronautics and Astronautics,
Purdue University,
701 W. Stadium Avenue, ARMS 3300,
West Lafayette, IN 47907
e-mail: yangzhang@purdue.edu

Ming Gan

School of Aeronautics and Astronautics,
Purdue University,
701 W. Stadium Avenue, ARMS 3300,
West Lafayette, IN 47907
e-mail: ganm@purdue.edu

Vikas Tomar

Associate Professor
School of Aeronautics and Astronautics,
Purdue University,
701 W. Stadium Avenue, ARMS 3205,
West Lafayette, IN 47907
e-mail: tomar@purdue.edu

1Corresponding author.

Manuscript received April 13, 2014; final manuscript received July 7, 2014; published online August 19, 2014. Assoc. Editor: Hsiao-Ying Shadow Huang.

J. Nanotechnol. Eng. Med 5(2), 021003 (Aug 19, 2014) (11 pages) Paper No: NANO-14-1034; doi: 10.1115/1.4027989 History: Received April 13, 2014; Revised July 07, 2014

Abstract

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.

<>

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 μm) 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 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 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 (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. 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 Ar+ 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 μm to tens of μm [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] Display Formula

(1)$Kf=2PπdΔT$

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 $ΔT$ is the temperature increase of the laser heated spot. $ΔT$ is determined by Display Formula

(2)$ΔT=t¯-Ts$

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]: Display Formula

(3)$I(r)=2Pπr02exp(-2r2r02)$

where $r0$ = d/2 is the radius of the laser spot and P is the laser power.

The general heat transfer equation is given as Display Formula

(4)$∂2t(r,z)∂r2+1r∂t(r,z)∂r+∂2t(r,z)∂z2=0$

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 ($Φ$) direction (due to sample size being significantly bigger than laser spot size), $Φ$ does not appear in this equation. In order to achieve homogeneous boundary conditions, the temperature was normalized by using the relation $t(r,z)=T(r,z)-Ts$. 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: Display Formula

(5)$∂t(r,z)∂z=f(r)=-I(r)-ht(r,z)k=-2Pkπr02exp(-2r2r02)+hkt(r,z), when z=0, and$
Display Formula
(6)$t(r,z)→0,when(r2+z2)1/2→∞$

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 Display Formula

(7)$g(q)=2π∫0∞f(r)J0(2πqr)rdr and$
Display Formula
(8)$f(r)=2π∫0∞g(q)J0(2πqr)qdq$

where $J0(2πqr)$ is a zeroth-order Bessel function of the first kind. Some properties of the Hankel transform are listed in Table 2. Taking $H(t(r,z))=Γ(q,z)$ and Hankel transform, Eq. (4) becomes Display Formula

(9)$d2Γ(q,z)dz2-q2Γ(q,z)=0$

After applying boundary conditions Eqs. (5) and (6) become Eqs. (10) and (11) shown below, respectively Display Formula

(10)$dΓ(q,z)dz=-P2πkexp(-r02q28)+hkΓ(q,z),whenz=0$
Display Formula
(11)$limz→∞Γ(q,z)=0$

The solution of Eq. (9) is in the form of Eq. (12)Display Formula

(12)$Γ(q,z)=A(q)exp(qz)+B(q)exp(-qz)$

Substituting the boundary conditions to Eq. (12) from Eqs. (10) and (11), the results are Display Formula

(13)$A(q)=0$
Display Formula
(14)$B(q)=P2πk(q+hk)exp(-r02q28)$

Thus Display Formula

(15)$Γ(q,z)=P2πk(q+hk)exp(-qz-r02q28)$

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

(16)$t(r,0)=2π∫0∞Γ(q,0)J0(2πqr)qdq$

where $I0(r)$ is the zero-order modified Bessel function of the first kind. The average temperature at the laser spot is Display Formula

(17)$t¯=1πr2∫0r0t(r,0)2πrdr$

Substituting Eqs. (15) and (16) in Eq. (17), the average temperature at the laser spot turns to be Display Formula

(18)$t¯=1πr02∫0r0t(r,0)2πrdr=2r02∫0∞2πqΓ(q,0)[∫0r0J0(2πqr)rdr]dq=2r0∫0∞Γ(q,0)J1(2πqr0)dq=Pπkr0∫0∞1q+hkexp(-r02q28)J1(2πqr0)dq$

$I1(r)$ 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 $cm-1$ and 980 $cm-1$. The Raman shift $Δω$ can be expressed in terms of wavenumber as, Display Formula

(19)$Δω=(1λ0-1λ1)$

Here, $λ0$ is the excitation wavelength, and $λ1$ 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 (Ar+ 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 Display Formula

(20)$d=λ/(πNA)$

where $λ$ 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 $exp(-2x2/r02)$ 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 24.5 μm. For most of our measurements, the laser spot size is found to be in the range of 19.13 μm–24.5 μm. 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 + 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: Display Formula

(21)$I1=I2+I3+Id$
Display Formula
(22)$I3=I4+Io$
Display Formula
(23)$I4=I4'+Is$
Display Formula
(24)$I3'=I4I3I4'$
Display Formula
(25)$I5=I2+I3I1I3'$

In these equations, $I1$$I5$ are the intensity of laser at the five points, respectively. $I3'$ and $I4'$ are the intensity of reflected laser at points 3 and 4 shown in Fig. 2(b). $Id$, $Io$, and $Is$ are absorbed power of laser by dichroic mirror, the objective, and the samples, respectively. In this part, only $I1$$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 $cm-1$ and 980 $cm-1$). 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 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 $cm-1$, 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 Display Formula

(26)$Δω=960.984-0.00873T$

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 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 relationships at the room temperature for the two samples are expressed, respectively, as

(29)$Δω=958.724-2.350×10-6|σ| for sample 3$

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.

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] Display Formula

(30)$RaL=gβ(Ts-T∞)L3να=gβt¯L3να$

The natural heat convection coefficient can be calculated as Display Formula

(31)$N¯uL=0.68+0.670RaL1/4[1+(0.492/Pr)9/16]4/9$
Display Formula
(32)$h¯=N¯uLkairL$

In the above equations, g is acceleration of gravity; $β$ is thermal expansion coefficient of air; L is the height of the sample; $ν$ is the kinematic viscosity of air; $α$ 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 Display Formula

(33)$h¯=N¯uLkairL=9.01+6.2302t¯1/4$

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 Display Formula

(34)$k=0.411+1.49×10-6|σ|$

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 Display Formula

(35)$k=0.458+4.86×10-7|σ|$
Display Formula
(36)$k=0.444+1.47×10-6|σ|$

Figure 10(c) plots the overall linear relation between thermal conductivity and compressive stress from 0 to 0.2 MPa, which is expressed as Display Formula

(37)$k=0.426+3.91×10-7|σ|$

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 Display Formula

(38)$k=0.373+1.67×10-6|σ|$

For the stress range of 0.1–0.2 MPa, the linear relation is expressed as Display Formula

(39)$k=0.652-9.22×10-7|σ|$

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 $cm-1$ 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 (<20 ms). The measured laser line was 514.52 nm, instead of 514.5 nm. This 0.02 nm difference leads to 0.755 $cm-1$ difference in the corresponding wavenumber, which is twice the wavenumber resolution of the system.

Exposure Time Effect.

During Raman spectroscopy measurements, higher laser power leads to stronger Raman shift signal and thus less exposure time. Short exposure time eliminates the background noise and effect of instability of the system. However, higher laser power can also lead to higher measurement error. In addition, the Raman shift is affected by temperature. When the laser power is higher than the threshold value, it creates detectable temperature increase on the sample surface, which results in Raman shift data mixed with temperature-induced part. Therefore, high laser power does not always increase the accuracy of the measurement. The ideal laser power is the threshold laser power.

Value of Thermal Conductivity.

According to Fig. 9, with the temperature increase, thermal conductivity of cortical bone decreases. Thermal transport occurs due to the migration of free electrons and lattice vibration waves. In pure metals, the electron contribution to conduction heat transfer dominates, while in nonconductors and semiconductors, the vibration of lattice is dominant [48]. The cortical bone belongs to the second aspect with the vibration of lattice dominant in heat transfer. With the temperature increase, the molecular motion and lattice vibration amplitude and frequency increase at the same time. Therefore, the increase in temperature should facilitate rapid thermal motion and higher thermal conductivity. However, increase in temperature also leads to increased interaction between various heat transfer modes. Such interaction has been shown to lead to reduction in thermal conductivity with increase in temperature. The presence of interfaces further facilitates such decrease. Some authors have earlier presented models that incorporate the role played by hierarchical interfaces in affecting thermal conductivity. Samvedi and Tomar [49] have earlier presented a model that accounts for biomimetic arrangement of interfaces in determining thermal conductivity of a set of Si-Ge biomimetic materials as a function of stress and temperature. The model does not include hierarchy of interfaces such as those present in cortical bone. Therefore, the model is not applicable directly here. However, qualitative conclusions from this work point out that biomimetic arrangement of interfaces leads to diffused phonon spectra owing to significant interface scattering. The scattering pattern follows a nonlinear trend due to the interfaces being present in a manner that does not promote direct increase in thermal boundary resistance in proportional to temperature increase (e.g., in superlattices all interfaces are arranged in direct path of thermal energy flow versus in a biomimetic material where those are arranged in an hierarchical fashion). Their analyzes showed that thermal conductivity indeed increases as a function of temperature in biomimetic materials as opposed to superlattice materials made up of the same components with the same volume fraction. The measurements reported in this work repeat those observations.

The trend in the case of applied compressive stress loading is not monotonic. Thermal conductivity has been shown to be dependent on the applied strain $ɛ$ as $k~T-1ɛ-γ$, where $γ$ is a material constant, $ɛ$ the strain value, and T is the temperature [50]. This expression implies that an increase in the thermal conductivity values when applying compressive strain. In the given setup of experiments, this only seems to be applicable in the low stress regime. In such low stress regime (<0.1 MPa) primarily the interface rotation and reorganization takes place. Therefore, the interfaces have minimal effect on thermal conductivity change. At higher stress values, the thermal conductivity variation shows different trend. One reason may be that the phonon modes increase coupling with each other with shortening of interatomic distances that follows the increase in applied stress. Hierarchical structure of bone may be another responsible factor. The compression leads to interfaces being closer to each other and, therefore, to increased blocking of phonon waves. One expression of thermal conductivity from kinetic gas theory is Display Formula

(40)$k=13CvsΛ$

where $k$ is the thermal conductivity; $C$ is the specific heat; $vs$ is the average phonon group velocity, and $Λ$ is the phonon mean free path (MFP). At higher stress values, compressed interfaces and formed microcracks contribute to significant reduction in phonon MFP that leads to reduction in thermal conductivity values beyond a threshold value (which is 0.1 MPa in this case).

Thermal Diffusivity.

In heat transfer analysis, the ratio of the thermal conductivity to the heat capacity is an important property termed the thermal diffusivity $α$, which is calculated by Display Formula

(41)$α=kρcp$

where $ρ$ is the density and $cp$ specific heat. It measures the ability of a material to conduct energy relative to its ability to store thermal energy. Materials of large $α$ will respond quickly to changes in their thermal environment, while material of small $α$ will respond more sluggishly, taking longer to reach a new thermal equilibrium condition.

According to Eq. (41), we can calculate thermal diffusivity of cortical bone after getting the thermal conductivity of it if the density and specific heat of cortical bone are considered to be constant. Huiskes [7] summarized the results of three investigators who measured the density of cortical bone. The range was $1.86×103$$2.91×103 kg/m3$, with an average of $2.21×103 kg/m3$. In the same research, Huiskes [7] summarized the published values for the specific heat of bone. While one researcher measured values in a range from $1.15×103$ to $1.73×103 J/(kg·K)$, two researchers reported the same value, $1.26×103 J/(kg·K)$. Our experimental results of thermal conductivity of cortical bone at room temperature are in range from 0.45 to 0.64 $W/(m·K)$, with an average of 0.54 $W/(m·K)$. So according to Eq. (41), the value of thermal diffusivity of cortical bone at room temperature is around $1.738×10-7m/s2$.

Compared to the metallic material or semiconductive material, the value of thermal diffusivity of cortical bone is much smaller, which means there will be much higher temperature at the laser spot that results from much more thermal energy aggregated here. That is why natural heat convection must be considered in Eqs. (5) and (10) about boundary conditions of heat transfer, as done in this work.

Conclusion

The experimental results of thermal conductivity of cortical bone at room temperature and at 0% strain are in the range from 0.45 to 0.64 $W/(m·K)$, with an average of 0.54 $W/(m·K)$. These values of thermal conductivity are in the range of the values reported by Moses et al. [12], Davidson and James [13], and Chato [15], which vary from 0.38 to 0.70 $W/(m·K)$. A summary of the discussed studies is presented in Table 1. The close match proves the effectiveness of the Raman spectroscopy method to measure the thermal conductivity of cortical bone. Analyzes established empirical relations between Raman shift and temperature as well as a relation between Raman shift and nanomechanical compressive stress. In addition, measured trends and established thermal conductivity–stress relations indicate that the thermal conductivity values increases and then decrease as a function of the compressive strain. Discussed mechanisms illustrate that phonon blocking is not effective in reducing thermal conductivity up to a certain threshold value (0.1 MPa) of stress. Below this value interfaces in the examined material play an insignificant role. With increase in stress values above the threshold value, the interfaces start dominating the stress–thermal conduction correlation.

Overall, as a nondestructive and noncontact thermal conductivity measurement method, Raman spectroscopy method is found to be suitable for thermal property measurements as a function of temperature and applied stress. Earlier, simulations on a biomimetic Si-Ge system have indicated increase in thermal conductivity with increase in temperature and reduction with increase in compressive stress above a threshold value. The reported work here confirmed such results for a real biological material through a new experimental measurement setup.

Acknowledgements

This work was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, under Award No. DE-SC0008619 (Program Manager: Dr. Michael Markowitz).

References

Jaasma, M. J., Bayraktar, H. H., Niebur, G. L., and Keaveny, T. M., 2002, “Biomechanical Effects of Intraspecimen Variations in Tissue Modulus for Trabecular Bone,” J. Biomech., 35(2), pp. 237–246. [PubMed]
Peterlik, H., Roschger, P., Klaushofer, K., and Frantzl, P., 2006, “From Brittle to Ductile Fracture of Bone,” Nature Mater., 5(1), pp. 52–55.
Thurnera, P. J., Ericksona, B., Jungmanna, R., Schriocka, Z., Weaverb, J. C., Fantnera, G. E., Schittera, G., Morseb, D. E., and Hansmaa, P. K., 2007, “High-Speed Photography of Compressed Human Trabecular Bone Correlates Whitening to Microscopic Damage,” Eng. Fract. Mech., 74(12), pp. 1928–1941.
Gan, M., Samvedi, V., Cerrone, A., Dubey, D. K., and Tomar, V., 2010, “Effect of Compressive Straining on Nanoindentation Elastic Modulus of Trabecular Bone,” Exp. Mech., 50(6), pp. 773–781.
Wiggins, K., and Malkin, S., 1976, “Drilling of Bone,” J. Biomech., 9(9), pp. 553–559. [PubMed]
Nelson, J. S., Yow, L., Liaw, L.-H., Macleay, L., Zavar, R. B., Orenstein, A., Wright, W. H., Andrews, J. J., and Berns, M. W., 1988, “Ablation of Bone and Methacrylate by a Prototype Mid-Infrared Erbium: YAG Laser,” Lasers Surg. Med., 8(5), pp. 494–500. [PubMed]
Huiskes, R., 1980, “Some Fundamental Aspects of Human Joint Replacement, Section II: Heat Generation and Conduction Analyses of Acrylic Bone Cement In Situ,” Acta Orthop. Scand., Suppl., 18, p. 43.
Zelenov, E., 1986, “Experimental Investigation of the Thermophysical Properties of Compact Bone,” Mech. Compos. Mater., 21(6), pp. 759–762.
Lundskog, J., 1972, “Heat and Bone Tissue. An Experimental Investigation of the Thermal Properties of Bone and Threshold Levels for Thermal Injury,” Scand. J. Plast. Reconstr. Surg., 9, pp. 1–80. [PubMed]
Vachon, R. I., Walker, F. J., Walker, D. F., and Nix, G. H., 1967, “In Vivo Determination of Thermal Conductivity of Bone Using the Thermal Comparator Technique,” Digest of the Seventh International Conference of Medical and Biological Engineering. Stockholm, Sweden.
Kirkland, R., 1967, “In Vivo Thermal Conductivity Values for Bovine and Caprine Osseous Tissue,” Proceedings of Annual Conference on Engineering in Medicine and Biology, Vol. 9, Boston, MA.
Moses, W. M., Witthaus, F. W., Hogan, H. A., and Laster, W. R., 1995, “Measurement of the Thermal Conductivity of Cortical Bone by an Inverse Technique,” Exp. Therm. Fluid Sci., 11(1), pp. 34–39.
Davidson, S. R., and James, D. F., 2000, “Measurement of Thermal Conductivity of Bovine Cortical Bone,” Med. Eng. Phys., 22(10), pp. 741–747. [PubMed]
Biyikli, S., Modest, M. F., and Tarr, R., 1986, “Measurements of Thermal Properties for Human Femora,” J. Biomed. Mater. Res., 20(9), pp. 1335–1345. [PubMed]
Chato, J., 1966, A Survey of Thermal Conductivity and Diffusivity Data on Biological Materials, ASME, ASME Paper 66-WA/HT-37.
Ribeiro, L. A., Rosolem, J. B., and Toledo, A. O., 2011, “Improving the Dynamic Range in Distributed Anti-Stokes Raman Thermometry by Means of Susceptibility Asymmetry,” 21st International Conference on Optical Fibre Sensors (OFS21), International Society for Optics and Photonics, Ottawa, Canada, Vol. 7753, p. 77532W.
Kim, S. H., Noh, J., Jeon, M. K., Kim, K. W., Lee, L. P., and Woo, S. I., 2006, “Micro-Raman Thermometry for Measuring the Temperature Distribution Inside the Microchannel of a Polymerase Chain Reaction Chip,” J. Micromech. Microeng., 16(3), pp. 526–530.
Phinney, L. M., Serrano, J. R., Piekos, E. S., Torczynski, J. R., Gallis, M. A., and Gorby, A. D., 2010, “Raman Thermometry Measurements and Thermal Simulations for MEMS Bridges at Pressures From 0.05 Torr to 625 Torr,” ASME J. Heat Transfer, 132(7), p. 072402.
Perichon, S., Lysenko, V., Remaki, B., Barbier, D., and Champagnon, B., 1999, “Measurement of Porous Silicon Thermal Conductivity by Micro-Raman Scattering,” J. Appl. Phys., 86(8), pp. 4700–4702.
Abel, M. R., and Graham, S., 2005, “Thermometry of Polycrystalline Silicon Structures Using Raman Spectroscopy,” ASME Paper No. IPACK2005-73088.
Gan, M., and Tomar, V., 2014, “An In Situ Platform for the Investigation of Raman Shift in Micro-Scale Silicon Structures as a Function of Mechanical Stress and Temperature Increase,” Rev. Sci. Instrum., 85(1), p. 013902. [PubMed]
Liu, X., Wu, X., and Ren, T., 2010, “A Metrology of Silicon Film Thermal Conductivity Using Micro-Raman Spectroscopy,” 2010 IEEE International SOI Conference (SOI), IEEE, San Diego, CA, pp. 1–2.
Dou, Y. W., Hu, M., Cui, M., and Zong, Y., 2005, “Experimental Study of Porous Silicon Thermal Conductivity Using Micro-Raman Spectroscopy,” The 6th International Symposium on Test and Measuremen t, Taiyuan, China, pp. 2094–2097.
Zhen-Qian, F., Ming, H., Wei, Z., and Xu-Rui, Z., 2008, “Micro-Raman Spectroscopic Investigation of the Thermal Conductivity of Oxidized Meso-Porous Silicon,” Acta Phys. Sinica, 57(1), pp. 103–110.
Pangilinan, G., and Gupta, Y., 1997, “Use of Time-Resolved Raman Scattering to Determine Temperatures in Shocked Carbon Tetrachloride,” J. Appl. Phys., 81(10), pp. 6662–6669.
Lo, H., and Compaan, A., 1980, “Raman Measurements of Temperature During CW Laser Heating of Silicon,” J. Appl. Phys., 51(3), pp. 1565–1568.
Reichling, M., Klotzbucher, T., and Hartmann, J., 1998, “Local Variation of Room-Temperature Thermal Conductivity in High-Quality Polycrystalline Diamond,” Appl. Phys. Lett., 73(6), pp. 756–758.
Poruba, A., Fejfar, A., Remes, Z., Springer, J., Vanecek, M., Kocka, J., Meier, J., Torres, P., and Shah, A., 2000, “Optical Absorption and Light Scattering in Microcrystalline Silicon Thin Films and Solar Cells,” J. Appl. Phys., 88(1), pp. 148–160.
Huang, S., Ruan, X., Fu, X., and Yang, H., 2009, “Measurement of the Thermal Transport Properties of Dielectric Thin Films Using the Micro-Raman Method,” J. Zhejiang Univ. Sci. A, 10(1), pp. 7–16.
Huang, S., Ruan, X., Zou, J., Fu, X., and Yang, H., 2009, “Thermal Conductivity Measurement of Submicrometer-Scale Silicon Dioxide Films by an Extended Micro-Raman Method,” Microsyst. Technol., 15(6), pp. 837–842.
Huang, S., Ruan, X., Zou, J., Fu, X., and Yang, H., 2009, “Raman Scattering Characterization of Transparent Thin Film for Thermal Conductivity Measurement,” J. Thermophys. Heat Transfer, 23(3), pp. 616–621.
Fan, Z., Swadener, J. G., Rho, J. Y., Roy, M. E., and Pharr, G. M., 2002, “Anisotropic Properties of Human Tibial Cortical Bone as Measured by Nanoindentation,” J. Orthop. Res., 20(4), pp. 806–810. [PubMed]
Rho, J.-Y., Tsui, T. Y., and Pharr, G. M., 1997, “Elastic Properties of Human Cortical and Trabecular Lamellar Bone Measured by Nanoindentation,” Biomaterials, 18(20), pp. 1325–1330. [PubMed]
Zysset, P. K., Guo, X. E., Hoffler, C. E., Moore, K. E., and Goldstein, S. A., 1999, “Elastic Modulus and Hardness of Cortical and Trabecular Bone Lamellae Measured by Nanoindentation in the Human Femur,” J. Biomech., 32(10), pp. 1005–1012. [PubMed]
Krause, W., 1987, “Orthogonal Bone Cutting: Saw Design and Operating Characteristics,” ASME J. Biomech. Eng., 109(3), pp. 263–271.
McDonald, F. A., and Wetsel, G. C., 1988, “Theory of Photothermal and Photoacoustic Effects in Condensed Matter,” Phys. Acoust., 18, pp. 167–277.
Ipatova, I., Maradudin, A., and Wallis, R., 1967, “Temperature Dependence of the Width of the Fundamental Lattice-Vibration Absorption Peak in Ionic Crystals. II. Approximate Numerical Results,” Phys. Rev., 155(3), pp. 882–895.
Wallis, R. F., Ipatova, I. P., and Aa, M., 1966, “Temperature Dependence of Width of Fundamental Lattice Vibration Absorption Peak in Ionic Crystals,” Soviet Physics Solid State, USSR, 8(4), pp. 850–861.
Rehman, I., Smith, R., Hench, L. L., and Bonfield, W., 1995, “Structural Evaluation of Human and Sheep Bone and Comparison With Synthetic Hydroxyapatite by FT-Raman Spectroscopy,” J. Biomed. Mater. Res., 29(10), pp. 1287–1294. [PubMed]
Pezzotti, G., and Sakakura, S., 2003, “Study of the Toughening Mechanisms in Bone and Biomimetic Hydroxyapatite Materials Using Raman Microprobe Spectroscopy,” J. Biomed. Mater. Res. Part A, 65(2), pp. 229–236.
Cai, W., Moore, A. L., Zhu, Y., Li, X., Chen, S., Shi, L., and Ruoff, R. S., 2010, “Thermal Transport in Suspended and Supported Monolayer Graphene Grown by Chemical Vapor Deposition,” Nano Lett., 10(5), pp. 1645–1651. [PubMed]
Morris, M. D., and Mandair, G. S., 2011, “Raman Assessment of Bone Quality,” Clin. Orthop. Relat. Res., 469(8), pp. 2160–2169. [PubMed]
Tchanque-Fossuo, C. N., Gong, B., Poushanchi, B., Donneys, A., Sarhaddi, D., Gallagher, K. K., Deshpande, S. S., Goldstein, S. A., Morris, M. D., and Buchman, S. R., 2012, “Raman Spectroscopy Demonstrates Amifostine Induced Preservation of Bone Mineralization Patterns in the Irradiated Murine Mandible,” Bone, 52(2), pp. 712–717. [PubMed]
De Wolf, I., 1996, “Micro-Raman Spectroscopy to Study Local Mechanical Stress in Silicon Integrated Circuits,” Semicond. Sci. Technol., 11(2), pp. 139–154.
Anastass, E., Pinczuk, A., Burstein, E., Pollak, F. H., and Cardona, M., 1970, “Effect of Static Uniaxial Stress on the Raman Spectrum of Silicon,” Solid State Commun., 8(2), pp. 133–138.
Incropera, F. P., Dewitt, D. P., Bergman, T. L., and Lavine, A. S., 2007, Fundamentals of Heat and Mass Transfer, 6th ed., John Wiley, Hoboken, NJ, p. 571.
Churchill, S. W., and Chu, H. H. S., 1975, “Correlating Equations for Laminar and Turbulent Free Convection From a Vertical Plate,” Int. J. Heat Mass Trans., 18(11), pp. 1323–1329.
Incropera, F. P., Dewitt, D. P., Bergman, T. L., and Lavine, A. S., 2007, Fundamentals of Heat and Mass Transfer, 6th ed., John Wiley, Hoboken, NJ, p. 61.
Samvedi, V., and Tomar, V., 2010, “Role of Straining and Morphology in Thermal Conductivity of a Set of Si-Ge Superlattices and Biomimetic Si-Ge Nanocomposites,” J. Phys.-D, Appl. Phys., 43(13), p. 135401.
Bhowmick, S., and Shenoy, V. B., 2006, “Effect of Strain on the Thermal Conductivity of Solids,” J. Chem. Phys., 125(16), p. 164513. [PubMed]
View article in PDF format.

References

Jaasma, M. J., Bayraktar, H. H., Niebur, G. L., and Keaveny, T. M., 2002, “Biomechanical Effects of Intraspecimen Variations in Tissue Modulus for Trabecular Bone,” J. Biomech., 35(2), pp. 237–246. [PubMed]
Peterlik, H., Roschger, P., Klaushofer, K., and Frantzl, P., 2006, “From Brittle to Ductile Fracture of Bone,” Nature Mater., 5(1), pp. 52–55.
Thurnera, P. J., Ericksona, B., Jungmanna, R., Schriocka, Z., Weaverb, J. C., Fantnera, G. E., Schittera, G., Morseb, D. E., and Hansmaa, P. K., 2007, “High-Speed Photography of Compressed Human Trabecular Bone Correlates Whitening to Microscopic Damage,” Eng. Fract. Mech., 74(12), pp. 1928–1941.
Gan, M., Samvedi, V., Cerrone, A., Dubey, D. K., and Tomar, V., 2010, “Effect of Compressive Straining on Nanoindentation Elastic Modulus of Trabecular Bone,” Exp. Mech., 50(6), pp. 773–781.
Wiggins, K., and Malkin, S., 1976, “Drilling of Bone,” J. Biomech., 9(9), pp. 553–559. [PubMed]
Nelson, J. S., Yow, L., Liaw, L.-H., Macleay, L., Zavar, R. B., Orenstein, A., Wright, W. H., Andrews, J. J., and Berns, M. W., 1988, “Ablation of Bone and Methacrylate by a Prototype Mid-Infrared Erbium: YAG Laser,” Lasers Surg. Med., 8(5), pp. 494–500. [PubMed]
Huiskes, R., 1980, “Some Fundamental Aspects of Human Joint Replacement, Section II: Heat Generation and Conduction Analyses of Acrylic Bone Cement In Situ,” Acta Orthop. Scand., Suppl., 18, p. 43.
Zelenov, E., 1986, “Experimental Investigation of the Thermophysical Properties of Compact Bone,” Mech. Compos. Mater., 21(6), pp. 759–762.
Lundskog, J., 1972, “Heat and Bone Tissue. An Experimental Investigation of the Thermal Properties of Bone and Threshold Levels for Thermal Injury,” Scand. J. Plast. Reconstr. Surg., 9, pp. 1–80. [PubMed]
Vachon, R. I., Walker, F. J., Walker, D. F., and Nix, G. H., 1967, “In Vivo Determination of Thermal Conductivity of Bone Using the Thermal Comparator Technique,” Digest of the Seventh International Conference of Medical and Biological Engineering. Stockholm, Sweden.
Kirkland, R., 1967, “In Vivo Thermal Conductivity Values for Bovine and Caprine Osseous Tissue,” Proceedings of Annual Conference on Engineering in Medicine and Biology, Vol. 9, Boston, MA.
Moses, W. M., Witthaus, F. W., Hogan, H. A., and Laster, W. R., 1995, “Measurement of the Thermal Conductivity of Cortical Bone by an Inverse Technique,” Exp. Therm. Fluid Sci., 11(1), pp. 34–39.
Davidson, S. R., and James, D. F., 2000, “Measurement of Thermal Conductivity of Bovine Cortical Bone,” Med. Eng. Phys., 22(10), pp. 741–747. [PubMed]
Biyikli, S., Modest, M. F., and Tarr, R., 1986, “Measurements of Thermal Properties for Human Femora,” J. Biomed. Mater. Res., 20(9), pp. 1335–1345. [PubMed]
Chato, J., 1966, A Survey of Thermal Conductivity and Diffusivity Data on Biological Materials, ASME, ASME Paper 66-WA/HT-37.
Ribeiro, L. A., Rosolem, J. B., and Toledo, A. O., 2011, “Improving the Dynamic Range in Distributed Anti-Stokes Raman Thermometry by Means of Susceptibility Asymmetry,” 21st International Conference on Optical Fibre Sensors (OFS21), International Society for Optics and Photonics, Ottawa, Canada, Vol. 7753, p. 77532W.
Kim, S. H., Noh, J., Jeon, M. K., Kim, K. W., Lee, L. P., and Woo, S. I., 2006, “Micro-Raman Thermometry for Measuring the Temperature Distribution Inside the Microchannel of a Polymerase Chain Reaction Chip,” J. Micromech. Microeng., 16(3), pp. 526–530.
Phinney, L. M., Serrano, J. R., Piekos, E. S., Torczynski, J. R., Gallis, M. A., and Gorby, A. D., 2010, “Raman Thermometry Measurements and Thermal Simulations for MEMS Bridges at Pressures From 0.05 Torr to 625 Torr,” ASME J. Heat Transfer, 132(7), p. 072402.
Perichon, S., Lysenko, V., Remaki, B., Barbier, D., and Champagnon, B., 1999, “Measurement of Porous Silicon Thermal Conductivity by Micro-Raman Scattering,” J. Appl. Phys., 86(8), pp. 4700–4702.
Abel, M. R., and Graham, S., 2005, “Thermometry of Polycrystalline Silicon Structures Using Raman Spectroscopy,” ASME Paper No. IPACK2005-73088.
Gan, M., and Tomar, V., 2014, “An In Situ Platform for the Investigation of Raman Shift in Micro-Scale Silicon Structures as a Function of Mechanical Stress and Temperature Increase,” Rev. Sci. Instrum., 85(1), p. 013902. [PubMed]
Liu, X., Wu, X., and Ren, T., 2010, “A Metrology of Silicon Film Thermal Conductivity Using Micro-Raman Spectroscopy,” 2010 IEEE International SOI Conference (SOI), IEEE, San Diego, CA, pp. 1–2.
Dou, Y. W., Hu, M., Cui, M., and Zong, Y., 2005, “Experimental Study of Porous Silicon Thermal Conductivity Using Micro-Raman Spectroscopy,” The 6th International Symposium on Test and Measuremen t, Taiyuan, China, pp. 2094–2097.
Zhen-Qian, F., Ming, H., Wei, Z., and Xu-Rui, Z., 2008, “Micro-Raman Spectroscopic Investigation of the Thermal Conductivity of Oxidized Meso-Porous Silicon,” Acta Phys. Sinica, 57(1), pp. 103–110.
Pangilinan, G., and Gupta, Y., 1997, “Use of Time-Resolved Raman Scattering to Determine Temperatures in Shocked Carbon Tetrachloride,” J. Appl. Phys., 81(10), pp. 6662–6669.
Lo, H., and Compaan, A., 1980, “Raman Measurements of Temperature During CW Laser Heating of Silicon,” J. Appl. Phys., 51(3), pp. 1565–1568.
Reichling, M., Klotzbucher, T., and Hartmann, J., 1998, “Local Variation of Room-Temperature Thermal Conductivity in High-Quality Polycrystalline Diamond,” Appl. Phys. Lett., 73(6), pp. 756–758.
Poruba, A., Fejfar, A., Remes, Z., Springer, J., Vanecek, M., Kocka, J., Meier, J., Torres, P., and Shah, A., 2000, “Optical Absorption and Light Scattering in Microcrystalline Silicon Thin Films and Solar Cells,” J. Appl. Phys., 88(1), pp. 148–160.
Huang, S., Ruan, X., Fu, X., and Yang, H., 2009, “Measurement of the Thermal Transport Properties of Dielectric Thin Films Using the Micro-Raman Method,” J. Zhejiang Univ. Sci. A, 10(1), pp. 7–16.
Huang, S., Ruan, X., Zou, J., Fu, X., and Yang, H., 2009, “Thermal Conductivity Measurement of Submicrometer-Scale Silicon Dioxide Films by an Extended Micro-Raman Method,” Microsyst. Technol., 15(6), pp. 837–842.
Huang, S., Ruan, X., Zou, J., Fu, X., and Yang, H., 2009, “Raman Scattering Characterization of Transparent Thin Film for Thermal Conductivity Measurement,” J. Thermophys. Heat Transfer, 23(3), pp. 616–621.
Fan, Z., Swadener, J. G., Rho, J. Y., Roy, M. E., and Pharr, G. M., 2002, “Anisotropic Properties of Human Tibial Cortical Bone as Measured by Nanoindentation,” J. Orthop. Res., 20(4), pp. 806–810. [PubMed]
Rho, J.-Y., Tsui, T. Y., and Pharr, G. M., 1997, “Elastic Properties of Human Cortical and Trabecular Lamellar Bone Measured by Nanoindentation,” Biomaterials, 18(20), pp. 1325–1330. [PubMed]
Zysset, P. K., Guo, X. E., Hoffler, C. E., Moore, K. E., and Goldstein, S. A., 1999, “Elastic Modulus and Hardness of Cortical and Trabecular Bone Lamellae Measured by Nanoindentation in the Human Femur,” J. Biomech., 32(10), pp. 1005–1012. [PubMed]
Krause, W., 1987, “Orthogonal Bone Cutting: Saw Design and Operating Characteristics,” ASME J. Biomech. Eng., 109(3), pp. 263–271.
McDonald, F. A., and Wetsel, G. C., 1988, “Theory of Photothermal and Photoacoustic Effects in Condensed Matter,” Phys. Acoust., 18, pp. 167–277.
Ipatova, I., Maradudin, A., and Wallis, R., 1967, “Temperature Dependence of the Width of the Fundamental Lattice-Vibration Absorption Peak in Ionic Crystals. II. Approximate Numerical Results,” Phys. Rev., 155(3), pp. 882–895.
Wallis, R. F., Ipatova, I. P., and Aa, M., 1966, “Temperature Dependence of Width of Fundamental Lattice Vibration Absorption Peak in Ionic Crystals,” Soviet Physics Solid State, USSR, 8(4), pp. 850–861.
Rehman, I., Smith, R., Hench, L. L., and Bonfield, W., 1995, “Structural Evaluation of Human and Sheep Bone and Comparison With Synthetic Hydroxyapatite by FT-Raman Spectroscopy,” J. Biomed. Mater. Res., 29(10), pp. 1287–1294. [PubMed]
Pezzotti, G., and Sakakura, S., 2003, “Study of the Toughening Mechanisms in Bone and Biomimetic Hydroxyapatite Materials Using Raman Microprobe Spectroscopy,” J. Biomed. Mater. Res. Part A, 65(2), pp. 229–236.
Cai, W., Moore, A. L., Zhu, Y., Li, X., Chen, S., Shi, L., and Ruoff, R. S., 2010, “Thermal Transport in Suspended and Supported Monolayer Graphene Grown by Chemical Vapor Deposition,” Nano Lett., 10(5), pp. 1645–1651. [PubMed]
Morris, M. D., and Mandair, G. S., 2011, “Raman Assessment of Bone Quality,” Clin. Orthop. Relat. Res., 469(8), pp. 2160–2169. [PubMed]
Tchanque-Fossuo, C. N., Gong, B., Poushanchi, B., Donneys, A., Sarhaddi, D., Gallagher, K. K., Deshpande, S. S., Goldstein, S. A., Morris, M. D., and Buchman, S. R., 2012, “Raman Spectroscopy Demonstrates Amifostine Induced Preservation of Bone Mineralization Patterns in the Irradiated Murine Mandible,” Bone, 52(2), pp. 712–717. [PubMed]
De Wolf, I., 1996, “Micro-Raman Spectroscopy to Study Local Mechanical Stress in Silicon Integrated Circuits,” Semicond. Sci. Technol., 11(2), pp. 139–154.
Anastass, E., Pinczuk, A., Burstein, E., Pollak, F. H., and Cardona, M., 1970, “Effect of Static Uniaxial Stress on the Raman Spectrum of Silicon,” Solid State Commun., 8(2), pp. 133–138.
Incropera, F. P., Dewitt, D. P., Bergman, T. L., and Lavine, A. S., 2007, Fundamentals of Heat and Mass Transfer, 6th ed., John Wiley, Hoboken, NJ, p. 571.
Churchill, S. W., and Chu, H. H. S., 1975, “Correlating Equations for Laminar and Turbulent Free Convection From a Vertical Plate,” Int. J. Heat Mass Trans., 18(11), pp. 1323–1329.
Incropera, F. P., Dewitt, D. P., Bergman, T. L., and Lavine, A. S., 2007, Fundamentals of Heat and Mass Transfer, 6th ed., John Wiley, Hoboken, NJ, p. 61.
Samvedi, V., and Tomar, V., 2010, “Role of Straining and Morphology in Thermal Conductivity of a Set of Si-Ge Superlattices and Biomimetic Si-Ge Nanocomposites,” J. Phys.-D, Appl. Phys., 43(13), p. 135401.
Bhowmick, S., and Shenoy, V. B., 2006, “Effect of Strain on the Thermal Conductivity of Solids,” J. Chem. Phys., 125(16), p. 164513. [PubMed]

Figures

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.

Fig. 2

(a) Overview of the experimental setup and (b) a schematic of the optical path of the experiments

Fig. 3

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

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.

Fig. 5

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

Fig. 6

Gauss fitting curve of Raman shift

Fig. 7

Correlation of temperature and Raman shift of cortical bone

Fig. 8

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

Fig. 9

Relation between thermal conductivity and temperature increase at the laser spot

Fig. 10

Thermal Conductivity of cortical bone (a) sample size 1 as a function of stress, (b) sample size 2 as a function of stress, and (c) sample size 3 as a function of stress

Tables

Table 1 Summary of comparable studies for bones
Table 2 Properties of Hankel transform

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related Proceedings Articles
Related eBook Content
Topic Collections