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Finite-element modeling of stressesand strains in a diamond anvil celldevice: case of a diamond-coatedrhenium gasketAndreiy I. Kondrat’yev a & Yogesh K. Vohra aa Department of Physics , University of Alabama at Birmingham ,Campbell Hall 310, 1300 University Blvd, Birmingham, AL35294-1170, USAPublished online: 22 Aug 2007.

To cite this article: Andreiy I. Kondrat’yev & Yogesh K. Vohra (2007) Finite-element modeling ofstresses and strains in a diamond anvil cell device: case of a diamond-coated rhenium gasket, HighPressure Research: An International Journal, 27:3, 321-331, DOI: 10.1080/08957950701557573

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High Pressure ResearchVol. 27, No. 3, September 2007, 321–331

Finite-element modeling of stresses and strains in a diamondanvil cell device: case of a diamond-coated rhenium gasket

ANDREIY I. KONDRAT’YEV and YOGESH K. VOHRA*

Department of Physics, University of Alabama at Birmingham, Campbell Hall 310, 1300 UniversityBlvd, Birmingham, AL 35294-1170, USA

(Received 22 March 2007; revised 29 May 2007; in final form 6 July 2007)

The stresses and strains in a diamond anvil cell device were investigated using a finite-element codeNIKE2D for the case of an ultra-hard composite gasket material.The pressure distribution in a diamond-coated rhenium gasket was measured by the energy dispersive diffraction technique to 213 GPa andcompared with the finite-element modeling results. We examine various models for the mechanicalproperties of diamond-coated rhenium gasket as well as for diamond failure for shear stresses exceed-ing 100 GPa. The elastic and plastic properties of gasket were varied such that a good agreementbetween the experimentally measured pressure distribution and the computational pressure profileswere obtained. As a result, we obtained the effective Young’s modulus, Poisson’s ratio, yield stress forindented gasket, linear hardening modulus, and hardening parameter value for this layered ultra-hardcomposite gasket material. Future diamond design strategies for attainment of extreme high pressuresusing ultra-hard gasket materials are also discussed.

Keywords: Diamond anvil cell device; Finite-element modeling; Diamond-coated rhenium gasket;Diamond anvil failure conditions

1. Introduction

Diamond anvil cell (DAC) device is the main tool for investigating the properties of materialsunder extreme conditions of 100–400 GPa pressures. DAC device components are: pusher,diamond anvils, sample chamber, and gasket. Finite-element modeling (FEM) and computersimulation are major methods for investigating properties of the compression process. FEMhas become one of the leading tools in computational mathematics and physics. FEM studieshave played an important part in high-pressure research as it allows for a theoretical opti-mization of diamond geometry leading to a generation of ultra high pressures in DAC device.This theoretical optimization is a preferred method over the trial and error experimentationduring optimization of DAC design, and FEM studies have led to a better understanding offailure mechanisms in DAC devices. Furthermore, FEM results also provide unique estimateof the shear stresses that are present in the interior of diamond anvil and gasket materialsas these quantities cannot be directly probed by the existing experimental techniques. We

*Corresponding author. Email: ykvohra@uab.edu

High Pressure ResearchISSN 0895-7959 print/ISSN 1477-2299 online © 2007 Taylor & Francis

http://www.tandf.co.uk/journalsDOI: 10.1080/08957950701557573

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322 A. I. Kondrat’yev and Y. K. Vohra

address these issues for the DAC compression experiment being made with a diamond-coatedrhenium gasket. We study the case of composite gaskets and as a special case consider a lay-ered structure of rhenium and diamond. It is proper to assume that the material properties aresomewhere in between those two boundaries, limiting model values: pure diamond and purerhenium. We suggest a new method, which allowed us to estimate the new material properties(diamond-coated rhenium gasket) compressed to the highest pressure of 213 GPa by FEMusing experimentally measured pressure distributions. The diamond deposition on rheniummetal was carried out by a microwave plasma chemical vapor deposition (CVD) process usinga methane/hydrogen gas mixture. It is possible to tailor the properties of this layered ultra-hardcomposite material by varying the thickness of the diamond layer.

We refer to earlier publications [1, 2] in FEM and computer simulation of DAC compressionand especially some relevant recent results [3, 4]. In earlier research, gasket materials suchas rhenium were considered and the yield strength of rhenium was modeled to be a linearlyincreasing function with increasing pressure [3, 4]. In these earlier FEM studies, ultra-hardcomposite materials containing diamonds have not been examined as possible gasket materials.In general, the use of ultra-hard diamond like gasket materials in a DAC device is expectedto result in a greater sample thickness at a given pressure when compared with conventionalgasket materials such as spring steel or pure rhenium. However, it is not clear at the presenttime whether the use of ultra-hard diamond-like gasket materials offer advantages in extendingthe working pressure range of a DAC device.

In our experiments, we have employed X-ray diffraction techniques at a synchrotron sourceto measure pressure distributions in a DAC device. The measured volume of hexagonal close-packed rhenium at a particular location in DAC was converted in to pressure using the roomtemperature equation of state [5, 6].

We study separately DAC device components: diamond anvils, gasket, and sample material.In those components, we study radial and axial stresses as well as pressure distribution withrespect to geometry of anvils, sample and gasket material properties, and radial and axialcoordinates. We have completed the description of geometry and material properties of all theDAC components. In some cases, diamonds were compressed to failure to test the upper limitof the pressures that can be created in DAC device. The most direct measurement that can beperformed is the radial pressure distribution in a DAC device. The pressure distribution in DACcan be experimentally measured only at the diamond/gasket interface and diamond/sampleinterface at z = 0. Obtaining experimental data of pressure vs. axial distance are more com-plicated problem. At this stage of DAC device technology, we are not able to measure axialand radial stresses. In order to calculate this information, we have to use FEM and computersimulation as well as analytical modeling and solution to main problems. For gasket and sam-ple material, the main parameter for describing material model which was used for computersimulation and analysis is yield strength. We used the yield strength for description of materialproperties of gasket and sample material. We consider that the sample yield stress is a linearfunction of pressure [4, 7],

σy = σ0y + kP. (1)

For example for rhenium, the linear relation for yield stress (GPa) will be [4, 8]

σy = 8.0 + 0.04P.

In our research, we used NIKE2D software system [9]. Geometry of a problem wasdescribed by the size of the central flat of diamond anvil and the value of the first bevelangle (other parameters including second bevel angle, size of a culet we considered to befixed). Diamond anvil was described as material type 1, elastic using the following parame-ters: density,Young’s modulus, and Poisson’s ratio. Gasket and sample material were described

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FEM of stresses and strains in diamond anvil cell device 323

as material type 3, kinematic/isotropic elasto-plastic. The hardening parameter β gives kine-matic behavior, β = 0, or isotropic behavior, β = 1, or a combination of both, if 0 ≺ β ≺ 1.The linear hardening law is given by the following formula

σy = σ0y + β · EP · εP, (2)

where σy is the current yield stress, εP is effective plastic strain, and Ep is the hardeningmodulus represented by a slope of the yield stress σy vs. effective plastic εP

εP =∫ t

0dεP, dεP =

(2

3dεP

ij dεPij

),

where dεPij is a plastic stress tensor. For isotropic hardening, the effective stress σ is

σ =(

3

2sij sij

)1/2

,

where sij is a deviatoric stress tensor. For kinematic hardening process, we have

σ =(

3

2ηij ηij

)1/2

,

where ηij is a translated stress tensor.

ηij = sij − αij

and αij is a back stress tensor. The tangent modulus ET is related to the elastic modulus (E)and the hardening modulus (Ep) by the following

Ep = E · ET

E − ET.

In this paper, we plan to (1) describe methods of measuring pressure distribution obtainedin DAC compression experiments, (2) FEM of DAC compression, (3) results of FEM andcomputer simulation.

2. Measuring pressure distribution in DAC device

2.1 X-ray diffraction techniques under high pressures

There are two basic diffraction techniques which are employed under high pressure in DACat a synchrotron source to measure pressure distributions. These are energy dispersive X-raydiffraction and angle dispersive X-ray diffraction techniques. We have used the energy dis-persive X-ray diffraction at the X-17C beam line, National Synchrotron Light Source at theBrookhaven National Laboratory for measuring pressure distribution.

Figure 1 shows the photomicrograph of a typical diamond-coated rhenium gasket witha sample hole of 50 μm in diameter. In this experiment, the microcrystalline diamond filmwas deposited by a microwave plasma vapor deposition method using methane/hydrogenchemistry. In figure 1, the diamond deposition was carried out after pre-indentation and afterthe sample hole has been prepared by electric discharge machining. Figure 1 also illustratesthat a uniform microcrystalline diamond film can be deposited on rhenium metal and that the

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324 A. I. Kondrat’yev and Y. K. Vohra

Figure 1. The photomicrograph of a diamond-coated rhenium gasket. The flat size of the diamond is 300 μm indiameter and the sample-hole size of 50 μm in diameter.A well-adhered microcrystalline diamond coating of rheniumgasket is achieved using microwave plasma chemical vapor deposition technique.

diamond film is well adhered to the underlying metal. Figure 2 shows the thin film X-raydiffraction pattern of diamond-coated rhenium gasket where the diffraction peaks from thedeposited cubic diamond and the original hexagonal close packed rhenium metal are clearlyidentified. In our experiment, a beveled diamond with 15 μm diameter tip with a bevel angle

Figure 2. Thin film X-ray diffraction spectrum of a diamond-coated gasket. The X-ray beam is incident at a grazingangle of 2◦ to the gasket surface. The diffraction peaks form the cubic diamond phase and from the hexagonal closepacked rhenium are clearly identified.

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FEM of stresses and strains in diamond anvil cell device 325

Figure 3. The experimentally measured pressure distributions at a peak pressure of 213 GPa obtained by theX-ray diffraction technique (X and Y are in μm). The gasket center is located at the point (0, 0), −15 ≤ X ≤ 15,

−150 ≤ Y ≤ 150.

of 9.5◦, and a culet of 350 μm was used. A rhenium gasket of 200 μm initial thickness waspre-indented to a thickness of 20 μm and then coated with a 2 μm thick CVD layer on oneside of gasket material. Since rhenium gasket serves as a pressure marker in this experiment,no sample-hole drilling is required. A maximum pressure of 213 GPa was achieved with thisdiamond-coated rhenium gasket. An increase in pressure beyond 213 GPa in this experimentresulted in a diamond failure. We compared the‘ pressure distributions calculated on NIKE2Dsimulation model using different material models with the experimental measured pressuredistribution on diamond coated rhenium gasket.

2.2 Experimentally measured pressure distribution

The main goal of FEM and computer simulation was using geometry and material descrip-tions obtained from the experiment develop a model whose outcome for pressure distribution is‘close’ to the one obtained from the experiment. In the X-ray diffraction geometry employed, a10 × 10 μm2 collimated polychromatic X-ray beam is incident along the Z-axis and is perpen-dicular to the X–Y plane of the gasket. The pressure distribution was obtained by translatingthe diamond cell in the X–Y plane and recoding the energy dispersive X-ray diffraction datafrom rhenium at different locations. The measured volume of rhenium at various locations wasconverted into pressure using the equation of state of rhenium metal. Figure 3 shows the fourexperimentally measured pressure distribution curves for diamond-coated rhenium gasket.It is to be noted that a full pressure profile was obtained along the Y -direction. However, dueto time constraints at the synchrotron facility, only a partial X-direction scan was done. Weused the Birch’s equation of state [5, 6] for rhenium and also for diamond-coated rheniumgasket to calculate the pressure distribution.

3. Finite-element model

3.1 Geometry model

Because of radial and axial symmetry, we consider that mesh assembly is defined from onlyupper and right part of DAC. In figure 4, main components of DAC geometry design for

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326 A. I. Kondrat’yev and Y. K. Vohra

Figure 4. The geometry of the diamond anvil, gasket, and pusher used in the FEM calculations.

NIKE2D computer code are shown. Sample chamber occupies one-third part of the diamondtip and is located in the diamond tip center. Geometry of a pusher is not important; its materialproperties are defined in the way that pusher is incompressible. The DAC geometry is describedonly by one-fourth size of the real DAC (figure 4). In our geometry model, we do not describelower part of DAC at all. Instead we present radial axis (z = 0) as steady and immoveable, andalso describe the motion of each DAC component in negative z-direction by defining initialaxial velocity for each component. As the result, DAC is presented only by the upper partand with certain mesh corresponding to each component. When motion is applied to each ofDAC components (defined by initial axial velocity and number of time-steps) DAC becomescompressed and mesh for each component changes differently.

3.2 Material model

Diamond anvil and ‘pusher’ are considered to be ‘elastic’ and each of them is described bydensity, Young’s modulus, and Poisson’s ratio. This model describes isotropic, linear elasticmaterial behavior. Gasket and sample material are considered to be ‘kinematic/isotropicelastic–plastic material’ and described by density, Young’s modulus, Poisson’s ratio, yieldstress, hardening modulus, and hardening parameter.

Gasket represents a composite, main part of which is rhenium (pre-indented to 20-μmthickness) and diamond coating (2-μm thickness). The schematic of the geometry is shownin figure 4.

The rhenium and diamond-coated rhenium gasket were described by E,Young’s modulus; ν,Poisson’s ratio; ρ, material density; σy , yield stress; EP, hardening modulus, and β, hardeningparameter. The anvil was described byYoung’s modulus, Poisson’s ratio, material density withthe following values: Ediamond = 1050 GPa, νdiamond = 0.1, ρdiamond = 3.5 g/cm3. The pusherwas described byYoung’s modulus, Poisson’s ratio, material density with the following values:Epusher = 100,000 GPa, ν = 0.45, ρ = 1014 g/cm3. As can be seen from the data, a pusherrepresents an artificial superhard material which is practically non-compressible.

In cylindrical coordinate system inbuilt in NIKE2D let stress tensor components be

↔σ =

⎛⎝σr 0 0

0 σθ 00 0 σz

⎞⎠, σr = σθ , r = 0, Along the z-axis.

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FEM of stresses and strains in diamond anvil cell device 327

We call σr the radial stress component and σz as the axial stress component.Pressure is defined in the following way using these components P = (2σr + σz)/3.

Maximum shear stress is also defined using these components as σmax = (σz − σr)/2.

3.2.1 Description of gasket indentation. Initial gasket thickness was 200 μm, and afterindentation the thickness became 20 μm and thus effective strain ε = −0.9 (in future we skipnegative sign).

We used the following initial values for this stage of compression [8]:

Erhenium = 463 GPa, Ediamond = 1050 GPa,

νrhenium = 0.49, νdiamond = 0.1,

ρrhenium = 21 g/cm3, ρdiamond = 3.5 g/cm3.

We assumed that gasket while being indented was under the final indentation pressure of100 GPa. The effective yield stress for gasket σy,eff was defined as the one obtained at a pressureof 100 GPa. Thus for rhenium, we have σy,ind(Rh) = σ0y + kP = 8 + 0.04 × 100 = 12 GPa.The yield stress of diamond was assumed to be constant at 35 GPa.

Figure 5 shows the effective strain–stress linear hardening curve for diamond-coated rhe-nium gasket compression process. Notice that both cases of indentation as well as compressionof already indented gasket can be represented by this curve.

We use figure 5 to calculate values of tangent and hardening modulus for rhenium at this stageof gasket indentation. For this case, we have σ2 = 100 GPa, σ1y = 12, ε1 = σ1y/E = 0.0259,E = 463, ε2 = 0.9. Then we have (figure 5),

ET = σ2 − σ1y

ε2 − ε1= 101 GPa, EP = 129 GPa.

Notice also that the gasket density did not change after the pre-indentation process. Also, thediamond deposition on a gasket does not change much the density value because it occupiesonly a small portion of a gasket. Hence, for us it is proper to consider density of a gasket tobe the same as for rhenium.

Figure 5. Stress–strain curves showing the various materials properties used in the FEM code. The stress–straincurves are shown for the two limiting cases of hardening parameters (β = 0 and β = 1).

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328 A. I. Kondrat’yev and Y. K. Vohra

3.2.2 Compression of indented gasket. At this stage we compress indented gasket andin order for us to pick up a correct material model values we looked at different combinationsof parameters defining process of compression. At this stage of compression, we again usedelasto-plastic material model for gasket. Thus, we have the following parameters defininggasket compression at this stage: Young’s modulus, Poisson’s ratio, yield stress, hardeningmodulus. We took hardening parameter β-value to be 0 for all ‘diamond like’ composites andto equal 1 for pure rhenium gasket. Each of these parameters should be considered to haveboundary values either close to diamond (as diamond or ‘diamond-like’) or close to rhenium(as rhenium).

(a) Case of ‘diamond-like’gasket: We consider that indented gasket was compressed from 10to 6 μm. Thus we have,

E = 1050, σoy = 35, ε0 = σ0y

E= 0.033, σ3 = 213, ε3 = 0.4,

ET = σ3 − σ0y

ε3 − ε0= 485, EP = E · ET

E − ET= 901.

All modulus values are in GPa. The last value of EP we used as upper limit. Notice thatthough this was diamond-based gasket, but because we used elasto-plastic model we hadto describe plastic deformation which we observed in experiment, notice that tangentmodulus value was close to one for rhenium Young’s modulus. Notice that in calculationsof hardening and tangent modulus, we did not use density and Poisson’s ratio. These valuesplay a separate role and we can observe their influence only after we run simulation model.

(b) Case of pure rhenium gasket: We consider that gasket was compressed from 10 μm to2 μm. For this case we have the following

E∗ = 463, σ1y = 12, ε1 = σ1y

E= 0.0259, σ2 = 213,

ε2 = 0.8, ET = σ2 − σ1y

ε2 − ε1= 260, EP = 593.

(c) Case of a ‘diamond like’-based composite with tuned parameters: The yield stress of allgasket materials is expected to increase with increasing pressure due to work hardeningand increase in elastic modulus with increasing pressure. In earlier research papers [3, 4]for pure rhenium, the yield stress at high pressure was approximated by σ(P ) = σ0y + kPwith k values ranging from 0.04 to 0.1 were considered for various hardening processes.Wehave taken a similar approach for our composite gaskets and treated hardening parameterβ and tangent modulus ET as free parameters.

Our final refinement based on experimental data gives the following:

E = 1010, ET ≈ 0.2E = 202, EP = 252.5 GPa, and β = 0.2.

In figure 6, we compare the experimental pressure distribution measured to 213 GPa andcompare it with FEM results for various materials model. The pure rhenium model does notfit the width of the pressure distribution and tends to produce a sharper pressure distributionthan the one that is observed experimentally. The pure diamond fit can be improved using theNIKE2D tuned model where a high-elastic stiffness is maintained and a hardening parameterβ of 0.2 is utilized with Poisson ratio close to that of pure rhenium.

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FEM of stresses and strains in diamond anvil cell device 329

Figure 6. Various calculated pressure distributions using the NIKE2D are compared with the experimental pressureprofile at a peak pressure of 213 GPa. The radial distance is in μm. The NIKE2D tuned model gives a better fit thanthe NIKE2D diamond model or a NIKE2D rhenium model.

4. Results of FEM and computer simulation

4.1 Diamond anvil failure

The diamond anvil failed when the pressure on anvil/gasket interface reached 213 GPa. Thecondition for diamond anvil failure is considered as the one when the maximum shear stressreaches value around 100 GPa at certain point on z-axis [4]. We represent these conditions inthe following way.

Along z-axis, r = 0 and let z∗ = zfailure, z0 = zinterface,

σmax(z∗) = σz(z

∗) − σr(z∗)

2∼= 100 GPa, P (z0) = 2σr(z0) + σz(z0)

3= 213 GPa.

To account for the experimentally observed fracture of diamond anvil at 213 GPa for thediamond-coated rhenium gasket, we have calculated shear stress in the interior of diamondanvil for the various material models considered in this paper. The shear stress as a functionof the axial coordinate is plotted for the pure rhenium, pure diamond and the optimizedor tuned model in figure 7. It is interesting to point out that at the observed failure point,the calculated shear stress is in the 110 GPa range. The calculated shear stress is the lowestfor the pure rhenium model and is the highest for the NIKE2D tuned model. This resultindicated that the use of ultra hard gaskets tend to increase the shear stress in the interiorof the diamond anvil and the maximum in shear stress tend to shift to lower z-values withincreasing strength of the gasket material. It has been noted in earlier FEM studies [4] thatincreasing the bevel angle beyond 8.5◦ generally used for beveled diamond anvils wouldlower the shear stresses in the interior of diamond anvil. However, the use of higher bevelangle leads to high level of plastic deformation in conventional gasket materials thus givingrise to mechanical instabilities. One application of the ultra-hard diamond like composite

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330 A. I. Kondrat’yev and Y. K. Vohra

Figure 7. The calculated maximum shear stress as a function of axial distance in μm for a diamond-coated rheniumgasket. The plotted curves are for the three different material models that were investigated and all correspond to thesame sample pressure of 213 GPa.

gaskets would be to employ them with high-bevel angles of 10◦–15◦ and reduce the high-plastic deformation. The use of higher bevel angle in the range of 10–15◦ would also reducethe increased shear stresses that were observed in this paper with the ultra-hard gaskets. Thisproposed combination of high-bevel angle with ultra-hard gasket materials may lead to ageneration of extreme pressures in diamond anvil devices and should be explored in futureexperimentation in DAC devices.

5. Conclusions

In this paper, we have examined the use of ultra-hard composites as a gasket material in aDAC. A 2-μm thick microcrystalline layer was deposited on a pre-indented rhenium gasket,and pressure distribution was experimentally measured to 213 GPa. The FEM studies werecarried out for various materials models ranging from pure rhenium metal to diamond-likecomposites. It was observed that a good agreement between the experimentally measuredpressure distribution and the FEM can be obtained for the case where elastic stiffness is nearthe diamond value and the yield strength is optimized close to the rhenium value. An analysisof the calculated shear stresses shows that the shear stress in the interior of diamond anvilincreases with the use of ultra-hard gasket materials and a failure is predicted close to a peakpressure of 200 GPa. Further optimization with the ultra-hard composite gaskets would requirethe use of diamond design parameters that would lower the shear stresses and delay diamondfracture to even higher pressures.

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FEM of stresses and strains in diamond anvil cell device 331

Acknowledgements

We appreciate fruitful discussions and help via e-mail communications with Dr S. Weirand Dr S. Merkel. We used PC version of NIKE2D software provided by Dr S. Weir anddeveloped by Dr M. Murphy. This material is based upon work supported by the Depart-ment of Energy (DOE) – National Nuclear Security Administration (NNSA) under grantno. DE-FG52-06NA26168.

References[1] D.J. Dunstan, Rev. Sci. Instrum. 60 3789 (1989).[2] D.M. Adams and A.C. Shaw, J. Phys. D: Appl. Phys. 15 1609 (1981).[3] S. Merkel, R. Hemley and H.-K. Mao, Appl. Phys. Lett. 74 656 (1999).[4] S. Merkel, R. Hemley, H.-K. Mao and D. Teter, Paper preseted at Proceedings of the AIRAPT-17, edited by M.H.

Manghnani, W.J. Nellis and M.F. Nicol (Universities Press, Hyderabad, India, 2000), pp. 68–73.[5] F. Birch, Phys. Rev. 71 809 (1947).[6] Y.K. Vohra, S.J. Duclos and A.L. Ruoff, Phys. Rev. B36 9790 (1987).[7] A.L. Ruoff, J. Appl. Phys. 46 1389 (1975).[8] M.B. Bever, Encyclopedia of Materials Science and Engineering, Volume 8 (MIT Press, Cambridge).[9] J.O. Hallquist, NIKE2D – a vectorized, implicit, finite deformation, finite element code for analyzing the static

and dynamic response of 2-D solids, Law. Liv. Nat. Lab. Report UCID-19677 (1983).

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