Quantitative physical modeling of the effect of precipitates in the subgrain interior on the creep curve and service life of P91

ABSTRACT This work deals with a physically based creep model of the martensitic 9% Cr-steel P91. In particular, we quantitatively study the effect of precipitates located in the subgrain interior on the creep curve and the lifetime of the material. Our creep model is capable of simulating the microstructural evolution parallel to creep curves and has been further developed towards predicting time-to-rupture (TTR) diagrams. We demonstrate how the number density, size and shape of the precipitates in the subgrain interior affect the creep curve and thus the lifetime of the material. Three according parameter studies on the material P91 compare microstructures with/without precipitates and/or assuming specific shapes by introducing aspect ratios. These simulations can be used as a basis for further development of materials with regard to the precipitation phases.


Introduction
Modern thermal power plants with high efficiency contain components exposed to elevated temperatures and stresses.These components deform plastically by a slow rate due to the phenomenon of creep, which is limiting their service time significantly.In order to increase the service time, one has to understand the underlying physical processes causing these deformations.In our case, the material in question is P91, which belongs to the group of martensitic 9-12% Cr steels.
Martensitic 9-12% Cr-steels are state of the art materials for power plants due to their high creep strength and oxidation resistance [1].In order to increase the lifetime of the material (lower effective costs per operation year) and/or the service temperature (higher efficiency of the plant, lower CO 2 output), one has to further improve the creep strength [1,2].Unfortunately, experimental approaches to this task are extremely costly and time-consuming, since the critical degradations take place after 10+ operating years.Modelling approaches can assist in better understanding the results and shorten the experimental times.For this reason, we focus on improving modelling approaches.
Phenomenological models can mimic the creep deformation of materials (see e.g. the detailed summary of those model type by Holdsworth et al. [3]).This type of models describes the creep deformation by a set of analytical functions with fit parameters and can be applied quickly.Unfortunately, this type of model has two disadvantages: (i) the fit parameters can vary with applied temperature, stress, the material type and even the material heat and (ii) the model will not give insight into the very reasons for a good/bad creep behaviour of a specific material and thus cannot guide towards improving an existing material.
In contrast, physical models can provide this information; however, they are generally more challenging to be set up.These models consider the underlying creep mechanisms including the microstructure, interactions within the microstructure, the microstructural evolution during the creep process and, finally, the manifestation of these processes in terms of creep deformation (rate).These models also provide the opportunity to rate different starting microstructures and their effect on creep life, which is presumably the reason for different creep behaviour of various material badges of the same material type (apart from unintended variations of experimental parameters).
This work deals with an extension of such a physically based creep model, which has previously shown good results for the martensitic 9% Cr-steel P91 [4].The extension of the model includes a physically more accurate representation of the effect of precipitates in the subgrain interior that impede mobile dislocations in their gliding processes.We show the effect of the improvements in the model, the effect of the precipitates, as well as their influence on the lifetime of the material, and suggest potential further improvements for future models.

Creep model of the IMK research group
In order to account for the microstructural changes of the material, a microstructurally based physical dislocation creep model has been developed by our working group [4].Since the creep strain rate can be linked to the current mobile dislocation density and -velocity, creep curves and TTR diagrams can be deducted from the evolution of the microstructure and the specific interactions therein.
The core of our model has been developed by Riedlsperger [4], based on pre-works of Ghoniem [5], Yadav [6] and Basirat [7].Starting point of a simulation is the initial microstructure and precipitation data (e.g. from the software MatCalc [8]).The model considers various aspects of the microstructural evolution and interactions of mobile dislocations (density ρ m ), static dislocations (density ρ s ), boundary dislocations (density ρ b ), and subgrains (radius R sbg ).The creep strain rate _ ε is calculated based on the Orowan equation.Orowan's law [9] has been modified with respect to the velocity of mobile dislocations by now using an effective velocity v eff instead of the pure sliding velocity v g .Thus, the dislocation climb across precipitates in the subgrain interior can also be taken into account.
The potential microstructural interactions within a subgrain are depicted in Figure 1.Here, X denotes an annihilation of dislocations and arrows denote transformations.
In the creep model, the following interaction, generation, annihilation, and/or transformation processes are modelled [10]: Most of these mechanisms correspond to the original concept of Ghoniem [5].In addition, a number of terms have been adapted, refined and/or modified in the creep models [4,6,10].The current, most comprehensive version of the creep model is described in the book chapter [10].Compared to the previous fundamental model published in 2020 [4], the damage model has been significantly improved in the 2022 work [10] considering two aspects: (i) the description of cavitation damage was optimised (better tertiary creep region), and (ii) the model for damage due to precipitate coarsening was removed, since potential 'damage effects' due to coarsening are already implicitly included due to the effect of precipitates on the motion of mobile dislocations as well as on subgrain coarsening in point (g) above.
This adaptation of the model resulted in a better transition region between primary and secondary creep.
The creep strain rate _ ε, eq. ( 1), and the effective dislocation velocity v eff , eq. ( 2), can now be calculated as follows: In equation (1) b denotes the Burgers vector, ρ m represents the mobile dislocation density, v eff is the effective dislocation velocity, M is the Taylor factor and D cav denotes the cavitation damage factor.Meanwhile, in equation (2), v g represents the glide velocity and v c represents the climb velocity of dislocations.Equation (2) has been modified by Witzmann et al. [11] compared to sources [4] and [10] to take into account the impact of needle and plate-shaped particles.Instead of using precipitation radii r p,i and densities N V,i , the 'effective' precipitation radii r p,i,eff and densities N V,i,eff have been introduced.More details for this will be presented in the next section.

Morphology of the precipitates
Since precipitates can have different morphologies, a specific shape factor C(h) was introduced by Sonderegger et al. [12] in order to include different impacts of spherical, plate-shaped, as well as needleshaped precipitates with one single factor, leading to the calculation of an effective radius r p,eff and an effective precipitate density N v,eff [11].It should be noted here that this factor must be derived specifically for each crystal system of a matrix and the factor mentioned in this paper refers to bcc.
In the extension of the creep model in reference [11], the spherical precipitates in the subgrain interior are replaced by rotational ellipsoids, since these can also be used to describe needle-or plate-shaped precipitates.For rotational ellipsoids, the shape can be defined with the help of two parameters: with the semi-axes a i (b i =a i ) and c i .With these semi-axes, an aspect ratio h i can be defined according to eq. ( 3): An aspect ratio of h > 1 corresponds to a needleshaped precipitate, an aspect ratio of h < 1 to a plateshaped precipitate, and h = 1 describes a spherical precipitate.
The equations for the effective precipitate density N v,eff and the effective radius of precipitates r p,eff are given by eq. ( 4) and ( 5), and the associated parameter C(h) is provided in eq. ( 6) [12].
It should be noted that eq. ( 6) already considers all main variants of relative orientations between precipitates' rotational axes and the slip systems of the bcc crystal in a statistical manner.It should also be noted that eq. ( 4)-( 6) and also the effective dislocation velocity, eq. ( 2), only apply to precipitates within subgrains.The precipitates located at the subgrain boundaries are considered within the subgrain mobility and the Zener pinning effect.
A cross-section of an ellipsoidal precipitate (h = 4) within a glide plane is visualised in Figure 2(a).In Figure 2(b) a spherical precipitate with the same volume as the spheroidal precipitate is shown, and the multiplication of the radius from (b) with the specific shape factor C(h) to obtain the radius r p,eff is shown in Figure 2(c).
Figure 2 demonstrates the need of the parameter C (h): lengthy precipitates are more likely to intersect a slip plane compared to spherical particles of the same volume.In the case of an aspect ratio of h = 4, the probability for a particle to intersect a slip plane is increased by 43% (statistical mean value).We consider this effect by changing the number density N V into an effective number density N V,eff .In addition, the mobile dislocations have to climb higher distances for overcoming those particles (both statements do not necessarily hold for each individual interaction, but for statistically considering all potential relative orientations).For details of the derivation, the reader is referred to the original paper by Sonderegger et al. [12].

Material and experimental details
In the following simulations, the 9% Cr-steel P91 was used as material, since there is a large amount of literature and experimental data available for this material.In order to obtain the precipitate data in asreceived state as well during creep, simulations of the precipitation kinetics were carried out with the software package MatCalc 6.02 (based on the SFFK model [13]).For this, the thermodynamic database 'mc_fe_v2.060.tdb' and the mobility database 'mc_fe_v2.012.ddb' were used, as well as the chemical composition summarised in Table 1.
Microstructural input parameters for the simulation were taken from various literature sources, whereby more detailed information can be found in reference [4].Table 2 shows the starting values selected for the simulation of the material P91 at 650°C/70 MPa.Since not all parameters of the creep model are directly accessible experimentally, some have to be slightly optimised with the help of one single master creep curve.These are: the damage parameter A, the prefactor of the glide velocity a 1 , the source density β of mobile dislocations emitted from statics in the subgrain boundary walls, the pipe diffusion coefficient D vp , the Holt constant K c and the activation volume V r .In order to obtain optimal values for these parameters, an optimisation algorithm based on a simple gradient method was added into the existing Matlab code from Mergl [17].Nevertheless, these values should be viewed critically, although Krenmayr [18] provided some guidelines for the range of these parameters in P91.Furthermore, the activation volume V r can be roughly estimated with the help of relaxation data [4].
In Witzmann et al. [11], two different parameter studies have been carried out.In parameter study 1, the modelled creep curve according to the previous model [4] (with spherical precipitates in the subgrain interior) is tested against an experimental creep curve.Parameter study 2 tests the experimental curve against the improved model considering aspect ratios of precipitates.In order to simulate a realistic microstructure of P91, NbC and AlN were assumed to be spherical (see [19] for NbC in P91).No morphology data on AlN were available for the authors.However, their phase fraction is negligible.TEM images of P91 from Carrizo et al. (Figure 5(a) in reference [20]) show that VN are needle-shaped.This morphology for VN was also found by Hättestrand et al. (Figure 2(b) in reference [21]) in EFTEM images for P92.Another source -Aydogan et al. [22] -describes acicular vanadium carbo-nitrides for 12% chromium steels, which were investigated by STEM.VN is therefore considered to be acicular.For the sake of completeness, it should be mentioned that VN can also be plate-shaped [19,23].
Therefore, an aspect ratio of h = 4 was chosen for a needle-like VN and NbC & AlN were accordingly simulated as spherical precipitates with h = 1.

Results and discussion
In the following, three parameter studies are presented.Studies 1 and 2 demonstrate the effect of precipitates on individual creep curves and microstructural evolution, whereas study 3 demonstrates the impact of precipitates on TTR diagrams.It should be noted that some parts of the results of studies 1 and 2 have already been presented by Witzmann et al. [11].
The final set of simulation parameters for both parameter studies 1 & 2 is shown in Table 3 (Ω represents the atomic volume).It should be noted here that if the morphology of the precipitates is changed, only the parameters a 1 and A have to be adjusted [11].
The simulated creep curves for the two parameter studies, the experimental creep curve and the ± 19% scatter band (according to EN ISO 204) are shown in Figure 3(a) and (b).The experimental curve in Figure 3 and also in the following images represent an averaged curve from over 20 experiments taken from Krenmayr [18].It can be seen that in parameter study 1 -Figure 3(a) -especially the primary and secondary creep regions for spherical precipitates in the subgrain interior are excellently reproduced.The simulation of the tertiary region, on the other hand, deviates slightly from the experimental creep curve.In Figure 3(b) -parameter study 2 -all three regions of the simulated creep curve are almost identical with the experimental creep curve, which means on the one hand that a simulation with elongated VN maps the creep curve more realistically and furthermore, that a very good parameter choice was made.

General strategy
In order to demonstrate the influence of the precipitates inside the subgrain on the creep curve, the two parameter studies are compared using various different simulations.Both parameter studies include: In addition to the results presented in [11], we present the microstructural evolution of the matrix depending on different morphologies of the precipitates in the subgrain interior and also the effect of these precipitate shapes on the TTR diagram, which are shown in the following sections.

Parameter study 1
Figure 4 shows the simulated creep curves of parameter study 1 (see Table 3 for parameters), the experimental creep curve and the ± 19% scatter band (according to EN ISO 204).
As shown in Figures 4 and 3 If the lifetimes are compared, the lifetime of the experimental averaged creep curve [18] with 8503 h (at 6% elongation) is similar to the lifetime of the fitted curve for h = 1 with 9096 h and also to the creep curve, which has no precipitates in the subgrain interior (8706 h).Simulations of needle-shaped precipitates of the VN with h = 4 and h = 6, show a significantly higher lifetime compared to the result for spherical precipitates with 10,113 h and 12,030 h, respectively.For the creep curve simulations with plate-shaped VNs, the increase in lifetime is less pronounced than for needle-shaped precipitates.An aspect ratio of h=¼ corresponds to a lifetime of 9670 h and h=⅙ corresponds to a lifetime of 10,275 h.
Remarkable is the minimal difference in lifetime of spherical precipitates in contrast to the lifetime without precipitates in the subgrain-interior of only 4%.If   we assume that subgrain-interior precipitates have significant impact on the creep lifetime, this difference appears too low and suggests improper model assumptions (spherical particles).

Parameter study 2
Figure 5 shows the simulated creep curves of parameter study 2 (see Table 3 for parameters), the experimental creep curve and the ± 19% scatter band (according to EN ISO 204).
For parameter study 2, the deviation of the lifetimes (in %) of several different aspect ratios is shown in Table 4.The master creep curve with needle-shaped VN (h = 4) served as reference point.
As mentioned earlier, the master creep curve (h = 4) for parameter study 2 achieved an excellent agreement with the experimental creep curve.If no precipitates in the subgrain interior are considered, the creep curve of this simulation shows the lowest lifetime of all variants.The second lowest lifetime is obtained by a simulation with spherical precipitates.As in parameter study 1, needle-shaped VN precipitates lead to a stronger increase in creep life than plate-shaped precipitates -corresponding values can be found in Table 4.The differences between the individual simulations now appear much more realistic.For example, taking VN into account with an aspect ratio of h = 4 increases the service life (compared to the same simulation without VN) by 16%.An aspect ratio of h = 6 even results in a 38% longer creep life compared to the VN-free simulation.

Microstructural evolution for different morphologies of precipitates inside subgrains
In the following section, the influence of the morphologies of precipitates in the subgrain interior on the microstructural evolution is studied.Figure 6 shows the evolution of (a) mobile dislocations (density ρ m ), (b) static dislocations (density ρ s ), (c) boundary dislocations (density ρ b ), and (d) subgrains (radius R sbg ) over time.
For the study of the microstructural evolution, the simulation parameters from study 2 were used.As shown in Figure 6(a) and (b), precipitates in the subgrain interior have a significant influence on the mobile and static dislocation density.Up to a creep time of 250 h, only minimal differences are evident for the different variants.Differences start to become evident between 250 and 1000 h of creep.After 1000 h, the curves increase parallel in the same slope.
The boundary dislocation density (Figure 6(c)) is also very similar at the beginning for all variants of the creep simulation.The difference between the different variants increases more with ongoing creep time.It is worth to mention, that the creep curve without precipitates in the subgrain interior already seems to converge towards a value at 7000 h, whereas the curve with the lowest boundary dislocation density (h = 6) still continues to drop after 8000 h.
Figure 6(d) is particularly remarkable -here almost no deviation is visible for all variants.This confirms the new model that precipitates in the subgrain interior have hardly any influence on the subgrain radius.

Parameter study 3 -time-to-rupture diagram (TTR-Diagram)
Since the complete set of parameters is known for the parameter study 2, a TTR diagram can be reconstructed from the calculated creep curves.For this purpose, creep curves for the stresses from 60 MPa to 120 MPa were simulated without any additional alterations.All input parameter remained the same, only the stresses were varied in intervals of 10 MPa.The variation of applied stress changes the effective velocity in eq. ( 2), since both glide and climb velocity contain a stressdependent activation term -see Riedlsperger et al. [4].By contrast, a 1 , the pre-factor of the glide velocity in the modified Orowan's law, is regarded as stressindependent, however impacting the modelled rupture times most significantly amongst all the non-fixed parameters stated in Table 3, see also [16,18].The TTR diagram is obtained, when plotting the fracture times logarithmically against the stress, as it can be seen in Figure 7. Furthermore, it should be noticed that in the TTR diagram different shapes of VN (in the subgrain interior) were distinguished, as it was already done in the variations of the creep curves and the microstructural evolution.
To compare the simulated data with experimental fracture times, the ECCC datasheet for P91 [24] was used.For this purpose, the equation, which was obtained by the Manson-Haferd model from 2195 experimental points, was used to determine the rupture times for 650°C and the corresponding stresses.As can be seen in Figure 7, the reference curve (with h = 4 for VN) shows perfect agreement with the experimentally determined data of ECCC.The shape variations of the precipitates in the subgrain interior run parallel to the reference line.
Simulated rupture times react most sensitively to changes of the VN morphology from spherical to needle shape, whereas the effect of plate shapes on the TTR seems to be much less pronounced.It is particularly noteworthy that there is only a minimal difference for no precipitates in the subgrain interior or spherical precipitates in the subgrain interior.

Conclusion
The present work refines the existing dislocation creep model [4] and emphasises the special importance of the morphology of precipitates in the subgrain interior.Until 2022 - [11] -precipitates were always assumed to be spherical in the creep model of our working group.Since they can also be needle-or plateshaped in alloys, an existing side-model for the impact of various particle shapes [12] was implemented to better simulate the microstructure evolution.This is of particular relevance because the distance to be covered during climbing changes depending on the orientation and size of the precipitate.
In order to visualise the influence of spherical, needle-shaped or plate-shaped precipitates in the subgrain interior, parameter studies with different aspect ratios and simulations without precipitates in the subgrain interior were performed.Here it is evident, that not only the size of the precipitates or the number density but also the morphology (spherical, needle-shaped or plate-shaped) have a considerable influence on the creep curve and the fracture time.According to the original model [4] without any shape effects (parameter study 1), VNs in the subgrain interior extend the service life by only 4%, whereas according to the new model (parameter study 2) by 16%.Accordingly, the new model with the assumption of acicular VN provides an improved simulation of the creep curve and also of the microstructure.
To reasonably validate the results of the creep curves, direct experiments regarding precipitates in the subgrain interior would be needed.Nevertheless, these simulations appear to provide quite realistic values.
The findings obtained are summarised below.
(1) The existing dislocation creep model is refined & the particular importance of the morphology of the precipitates became apparent from the parameter studies (2) A more realistic microstructure can be simulated (3) Not only the size and number density of precipitates but also the morphology (needleshaped, plate-shaped) affects the creep curve (4) Partial explanation of different creep behaviour, when different batches of material are compared with each other (±20% scatter band related to stress) (5) The different morphologies of the precipitates in the subgrain interior affect the evolution of the mobile and static dislocation density (6) Creep curve and TTR -diagram can be reproduced very accurately (7) Our creep curve and TTR simulation demonstrated that needle-like precipitates in the subgrain interior lead to the most significant extensions of creep life (8) Almost no difference in the TTR diagram was visible whether spherical precipitates or no precipitates were assumed in the subgrain interior A possible further improvement is the implementation of lattice stresses into the model.Lattice stresses form around particles, as soon as the Youngs modulus of the particles differs from the matrix, and external forces are applied [25].Such stresses can potentially impact the glide and climb processes of dislocations near particles.This effect can increase the effective radius of particles and thus increase their impact on dislocation motion.Furthermore, the shape of the precipitates at the subgrain boundaries is currently not taken into account in the creep model.An extended model based on these precipitates located on subgrain boundaries would have an effect on the subgrain coarsening and thus improve the representation of the creep behaviour.The current model for needle and plate-shaped precipitates is planned to be implemented in the software CreeSo [26].
(a) Frank-Read sources (generation of mobile dislocations ρ m ) (b) Emission of mobile dislocations ρ m from static dislocations ρ s in the subgrain wall (c) Immobilisation of mobile dislocations ρ mgeneration of static dislocations ρ s (d) Recovery by climbing processes (=static recovery) for mobile and static dislocations.(e) Spontaneous annihilation (=dynamic recovery) of mobile and static dislocations (f) Subgrain boundaries are formed by static dislocations (g) Subgrain coarsening, subgrain nucleation and the Zener pinning effect of precipitates

Figure 2 .
Figure 2. Cross section of (a) an ellipsoidal precipitate with h=4; (b) a spherical precipitate with the same volume; (c) application of the parameter C(h) on the radius of the precipitate.

•
Simulations without precipitates (VN, NbC & AlN) in the subgrain interior.• Simulations with different variations of the aspect ratio h for VN.NbC and AlN are again considered to be spherical.It should be noted that h = 1 corresponds to a spherical precipitate, h < 1 to a plate-like precipitate, and h > 1 to a needle-like precipitate.• In each case, spherical chromium carbides M 23 C 6 are assumed to be located at the subgrain boundaries.
Figure4shows the simulated creep curves of parameter study 1 (see Table3for parameters), the experimental creep curve and the ± 19% scatter band (according to EN ISO 204).As shown in Figures4 and 3(a), the calculated creep curve for h = 1 (spherical) reproduces the primary and secondary creep stage excellently.The variant of the creep curve without precipitates in the subgrain interior is closer to the experimental creep curve in terms of the rupture time, but shows stronger deviations in the secondary creep stage compared to experimental data.If the lifetimes are compared, the lifetime of the experimental averaged creep curve[18] with 8503 h (at 6% elongation) is similar to the lifetime of the fitted curve for h = 1 with 9096 h and also to the creep curve, which has no precipitates in the subgrain interior (8706 h).Simulations of needle-shaped precipitates of the VN with h = 4 and h = 6, show a significantly higher lifetime compared to the result for spherical precipitates with 10,113 h and 12,030 h, respectively.For the creep curve simulations with plate-shaped VNs, the increase in lifetime is less pronounced than for needle-shaped precipitates.An aspect ratio of h=¼ corresponds to a lifetime of 9670 h and h=⅙ corresponds to a lifetime of 10,275 h.Remarkable is the minimal difference in lifetime of spherical precipitates in contrast to the lifetime without precipitates in the subgrain-interior of only 4%.If

Figure 4 .
Figure 4. Creep curves for different variations of the precipitates inside subgrains with the fitting parameters of the parameter study 1; *without precipitates = without VN, NbC & AlN inside subgrains (modified from[11]).

Figure 5 .
Figure 5. Creep curves for different variations of the precipitates inside subgrains with the fitting parameters of the parameter study 2; *without precipitates = without VN, NbC & AlN inside subgrains (modified from[11]).

Table 3 .
Final set of remaining simulation parameters of parameter study 1 & 2.