Active grain growth control with distributed heating

Active Grain Growth Control with Distributed Heating

Journal Pre-proof

Active Grain Growth Control with Distributed Heating Chengjian Zheng, Yixuan Tan, John T. Wen, Antoinette M. Maniatty PII: DOI: Reference:

S1359-6454(19)30676-7 https://doi.org/10.1016/j.actamat.2019.10.016 AM 15584

To appear in:

Acta Materialia

Received date: Revised date: Accepted date:

29 July 2019 2 October 2019 6 October 2019

Please cite this article as: Chengjian Zheng, Yixuan Tan, John T. Wen, Antoinette M. Maniatty, Active Grain Growth Control with Distributed Heating, Acta Materialia (2019), doi: https://doi.org/10.1016/j.actamat.2019.10.016

This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Graphical Abstract 140

grain size vs. time

120

z1 z2 z3 z4 z5 z6 z7 z8 z9 z10 zr

2

2

z =D (µ m )

100 80 60 40

Temperature Distribution

20 0

0

200

t=0 à 2000 min

400 600 Time(min)

800

16

grain size distribution

14

Grain Size D (µ m)

Grain Size Distribution

12

1000

t=0 t=100min t=500min t=1000min t=1500min t=2000min

10 8 6 4 2 0

Inner-Outer Loop Fbk

0

0.2

Output Feedback

MPC

0.4 0.6 X (mm)

0.8

Cu Ti heater array

SiO 2 Si

Temperature FEM Model

Grain Growth Model

1

1

Active Grain Growth Control with Distributed Heating Chengjian Zheng1 , Yixuan Tan2 , John T. Wen3 , Antoinette M. Maniatty4 Rensselaer Polytechnic Institute Troy, NY 12180

Abstract Microstructure affects the physical properties and behavior of materials. While metallurgists have long studied microstructure characterization and evolution, thermo-mechanical material processing to achieve a desired microstructure remains largely experience-based. This paper presents a distributed thermal control methodology for the microstructure evolution. We consider the problem of achieving a uniform microstructure, starting from a non-uniform initial distribution. This is a common goal in material processing, as uniform microstructure implies consistent macroscopic properties. To illustrate the approach, we consider an example process with a multi-zone micro-heater array controlling the grain growth of a copper thin film. Cascaded temperature and grain-growth models characterize the process dynamics – finite element method (FEM) models the temperature field in response to the heater input, which in turn drives the microstructure evolution through a biased Monte-Carlo (MC) model. The high order combined FEM/MC model is used as the validation “truth” model. For the control design and analysis, a simplified model is developed to only capture the essential trend in the full model. Using the simplified model and dividing the copper thin film into multiple spatial zones with mea1 Chengjian

Zheng

is

with

Bloomberg

Inc.,

New

York,

New

York.

[email protected] 2 Yixuan

Tan is with Google Inc., Mountain View, CA. [email protected] T. Wen is with the Electrical, Computer and Systems Engineering Department, Rensselaer Polytechnic Institute, Troy, New York. [email protected] 4 Antoinette M. Maniatty is with the Mechanical, Aerospace and Nuclear Engineering Department, Rensselaer Polytechnic Institute, Troy, New York. [email protected] 3 John

Preprint submitted to Elsevier

November 12, 2019

surable grain statistics in each zone, we obtain a nonlinear multi-input/multioutput control design model. Using the simplified model, this paper presents the development and comparison of three control methods: 1. Direct output feedback from the measured mean local grain sizes to the heater current. 2; Model predictive control (MPC) using a finite horizon optimization to compute the required heat input at each control step; 3. Inner-outer loop control with temperature as the surrogate input for the outer loop and using the heater current to achieve the required temperature in the inner loop. All three methods achieve uniform microstructure in grain growth in the higher order FEM/MC simulation. Direct output feedback is the simplest to implement, but has the slowest convergence. MPC shows fast convergence but requires model-dependent online optimization. Inner-outer loop demonstrates good compromise between model-dependence and rate of convergence. Keywords: Material Microstructure, Grain Growth, Monte Carlo Simulation, Consensus Control, Model Predictive Control

1. Introduction Microstructure is a crucial factor for the characterization and design of materials. It directly determines physical properties such as ductility, strength, electrical resistance [1]. Microstructure evolution takes place during thermalmechanical processing, e.g., cold work, hot rolling, heat treatment. Metallurgists have long focused on controlling microstructure to optimize material properties [2]. For example, Stubbington et al. improved the fatigue strength of Ti-6Al-4V by producing a microstructure with a small α grain size (< 10µm) with spheroidal β particles through a series of thermal mechanical processing steps [3]. Jahazi et al. studied the influences of hot rolling parameters on the microstructure and mechanical properties of steel [4]. Kirka et al. develop a heating strategy to control the grain structure, columnar versus equi-axed, for powder-bed fusion additively manufactured Inconel 718 [5]. These approaches adjust either processing conditions or process duration to achieve different final

3

microstructures. Results from multiple processing conditions are accumulated to infer the favorable operating condition. Analytical models of microstructure evolution have been developed to assist with process optimization. For example, Gama et al. investigated model-based optimization for microstructure design in the hot rolling process of stainless steel [6]. There have been efforts to connect processing parameters with microstructure through statistical methods [7]. These approaches are all open-loop which means the initial material state is assumed known and the microstructure evolution is assumed sufficiently accurate (either through design of experiment or computational models). Closed-loop microstructure control adjusts process inputs based on in-situ measurements of process variables. Such a control scheme is attractive as it does not require known initial material condition nor accurate model – the appropriate feedback of the deviation of the process measurements from the desired trajectory corrects for process modeling errors and disturbances. Feedback control requires in-situ measurements and their correlation to the microstructure and a microstructure evolution model to develop and analyze the controller and closed-loop performance. Control analysis and design usually uses a simplified reduced-complexity model. In [8], Gallivan et al. develops a feedback controller for a thin film growth process by using a reduced order model of the Kinetic Monte Carlo (KMC) process. In [9], optimal feedback control using dynamic programming based on a low dimensional reaction coordinate model is demonstrated for the assembly of colloidal crystals.

In [10], Lou et al.

models thin film surface roughness using stochastic partial differential equation (PDE), and applies model reduction for the controller design. In the control of batch crystallization process [11, 12, 13], deterministic nonlinear models are used to predict crystal size distribution, and these models are simplified for the control design. When direct microstructure measurements are not available, intermediate variables, such as temperature distribution, may be monitored and regulated instead. In semiconductor manufacturing, to achieve uniform wafer material properties, distributed temperature control has long been used in rapid thermal processing as a surrogate objective [14, 15]. 4

In Laser Additive Man-

ufacturing (LAM) [16, 17], microstructure is not directly measured, but a key processing condition that affects microstructure, the cooling rate, is estimated and regulated during run time. Grain growth is the most widely studied type of microstructure evolution [2]. In polycrystalline materials, grains are crystallites – regions with the same crystallographic orientations. During thermal treatment (e.g., annealing), grain boundaries – the interfaces between adjacent grains – migrate to decrease the stored energy in the system, resulting in small grains consumed by large grains and an increase in the average grain size. This process is known as grain growth. Experiments have suggested a direct relationship between annealing temperature and the average grain growth rate (higher temperature leads to faster growth). Studies on grain growth control have been experience-based and openloop, and for bulk materials rather than grain growth in different regions [2]. The goal for our work is to develop a systematic process control methodology to actively regulate distributed grain growth by adjusting the thermal processing condition. This approach may be used to achieve a uniform microstructure or to create functionally graded microstructures. As a case study, we consider the heat treatment of a copper thin film. The material is chosen because of its wide application in semiconductor devices, e.g., shaping its microstructure could enhance the etching factor [18]. The thin film is on top of a multi-zone heater array to spatially control the temperature distribution, and hence the rate and distribution of grain growth. Figure 1a illustrates a schematic cross-sectional view of the heater array design. Figure 1b shows a sample micro-fabricated heater array from our lab [19]. Ten titanium micro heaters are patterned underneath the copper thin film, enabling temperature manipulation across the horizontal direction. In our previous publications, an FEM model has been developed to simulate the temperature field evolution of the heater array [20] and a biased Monte-Carlo (MC) model has been developed to predict the evolution of grain structure under spatially-varying temperature field [21]. We assume the availability of real-time microstructure measurements (e.g., images from a Scanning Electron Microscope (SEM)) for the feedback control [22]. The con5

trol scenario is to start from a spatially non-uniform grain structure and adjust the heater input to achieve spatial uniformity based on grain size distribution feedback. This is similar to that of the consensus problems [23] – adjusting the action from multiple interacting agents to drive multiple process outputs to the same value. We consider the full FEM/MC simulation (combining results in [20, 21]) as the “truth” model, and use it for validation. For the controller design, we approximate the FEM model by a static map and grain growth by an empirical model for the average grain size in each spatial zone. In the simplified design model, the zones are coupled through the thermal dynamics, but the zone grain growth models are decoupled.

Two specific control objectives are

considered: consensus-driven grain growth which minimizes spatial grain size variations, and trajectory-driven grain growth which tracks a designed target grain growth trajectory. Three control strategies are developed for each control objective: direct output feedback, model predictive control (MPC), and inner-outer loop control. Direct output feedback uses the measured gain size distribution to directly adjust the heater current. MPC uses the thermal and grain growth models to optimize the heater current profile at regular intervals. Inner-outer loop approach uses the temperature field as a surrogate input in the inner loop and heater current in the outer loop to achieve the required temperature distribution. All three control methods demonstrate convergence in the FEM/MC validation simulation. Direct output feedback requires minimal model information (just some bounds) but the convergence is the slowest. MPC shows the fastest convergence but is computationally intensive and highly model-dependent. Inner-outer loop control offers a good compromise in terms of computation time, implementation complexity, model dependence, and convergence rate. A preliminary version of this paper [22] has presented early results of the output feedback and MPC methods in the simplified model. This paper includes the inner-outer loop method and the full order simulations. This paper consists of the following parts: Section 2 presents the details in heater array and grain growth modeling. Section 3 discusses control design and analysis. Section 4 summarizes the simulation results and discussions. 6

(a)

(b)

Figure 1: (a) Schematic cross-section view of the heater array design (b) Top view of heater array micro-fabricated at RPI. The orange area is overlaid onto the original image to show the position of copper thin film to be deposited.

2. Process Modeling 2.1. Heater Array Model The temperature evolution in the copper thin film in response to the micro heater array may be captured by nonlinear dynamic FEM simulation [20]. Neglecting temperature variation along the length of the heaters, the model captures the 2D temperature distribution. The thermal system is inherently stable and has fast dynamics (a consequence of the thin film), with time constant around 10ms. Significant grain growth only occurs in the time scale of tens of minutes. Hence, we ignore the temperature variations across the thickness of film and approximate the temperature field as a static map between the input current u and temperature field on the copper surface Ts : Ts = f (u),

(1)

where f (·) ∈ RNr → R` , Nr is the number of heater lines, ` is the number of nodes of interest on the copper surface. To further reduce the computational time of the thermal model, we use an artificial neural network (ANN) [24] to approximate (1). We find that an ANN 7

with one hidden layer and hyperbolic tangent sigmoid activation functions [25] is adequate to capture the nonlinearity in f (·) [20]. To train the ANN, simulations on the full-order FEM model is repeated for multiple input current profiles to generate the training dataset. Then we apply the Levenberg-Marquardt algorithm to learn the weights and biases to best match ANN prediction with the training data. 2.2. Grain Growth Model 2.2.1. Monte-Carlo Model We use the biased Monte Carlo(MC) model [21, 22] to model grain growth in a non-uniform temperature field. The details are summarized in Appendix A. The following widely-accepted exponential growth rule [26, 27] is used in the biased MC model to link grain growth to physical time: ¯ Dn − D0n = Kexp(−

Q )t RT

(2)

where D denotes the mean grain size, D0 is the initial grain size, t the physical ¯ the grain growth constant, Q the activation energy, R the Boltzmann time, K constant, T the temperature (in Kelvin). The grain growth exponent, n, is Q generally accepted as 2 for copper [27]. The Arrhenius term exp(− RT ) is related

with grain boundary mobility, which increases with T . 2.2.2. Multi-Region Empirical Model Most microstructure characterizations focus on the mean grain size. This suggests using (2) as the basis of a simplified grain growth model. To extend to a non-uniform temperature field, consider the domain as the union of multiple disjoint regions within which the temperature variation is negligible. In this case, (2) may be used to approximate the evolution of the mean grain size of the ith region, Di :

d(Din ) Q ¯ = Kexp(− ), dt Ryi

(3)

where yi denotes the average temperature in the region. This model ignores grain growth coupling at interfaces between regions, which is insignificant when 8

grain size is much smaller than the zone size. In addition, there are also other model errors from neglected temperature variation inside each region and grain size measurement. Using zi = Din , n = 2, we may approximate the average grain growth dynamics as: z˙ = g(y),

Q ¯ g(yi ) := Kexp(− ) Ryi

(4)

where z = [z1 , . . . , zNr ]T is the collection of average zone grain sizes, and y = [y1 , . . . , yNr ]T is the collection of average zone temperatures. The zone temperatures, y, may be obtained by averaging the temperature outputs in (1). We write the resulting relationship as y = F (u) where yi = Fi (u) =

R

x∈Xi

(5)

f (u) dx, Xi is the ith zone. Note that for our example

system, the domain is divided into 10 regions corresponding to 10 heaters, so Nr = 10.

3. Control Design 3.1. Control Problem Formulation The cascaded FEM and MC models shown in Figure 2a is used as the “truth” model to evaluate the controller performance. The simplified model used in the control design as shown in Figure 2b is based on a static temperature map (5) and the empirical grain growth model (4). The control objective is to choose u, 0 ≤ ui ≤ umax , to achieve both desired growth and uniformity. We may approach this objective from either of the following directions: A. Trajectory-based Control: Drive the average grain size in every zone to track a reference growth trajectory. More specifically, choose u to drive maxi |zi − z ∗ | → 0, where z ∗ is the grain growth trajectory corresponding to u∗ (i.e., z˙ ∗ = g(F (u∗ ))).

9

B. Consensus-based Control: Minimize the grain size error between Nr regions while achieving the desired growth rate. More specifically, let u = u∗ + u1 , where u∗ is designed for the desired consensus growth rate and u1 is chosen to drive maxij |zi − zj | → 0.

(a)

(b)

Figure 2: (a) Validation model structure (b) Control design model structure

We consider three candidate control strategies to meet the control objective: 1. Direct Output Feedback: The measured averaged grain sizes, z, is directly used to control the heater current, u, as shown in Figure 3a. Explicit model information is not needed in the controller implementation. The inherent passivity property of the system is used to show closed loop stability. However, the input constraint is not directly incorporated in the control design; it is only indirectly enforced through gain tuning. 2. Model Predictive Control (MPC). A finite horizon optimization based on the measured z and linearized system to solve for a future input trajectory, as shown in Figure 3b. The input constraint is directly included as part of the optimization problem. However, the dependence on the model for solving the optimization problem is susceptible to model uncertainty. This issue is particularly acute in the grain growth model due to unknown activation energy Q and conditions of specific thin film samples. 3. Inner-Outer Loop Control. This approach uses temperature as a surrogate input to control the grain growth outer loop, and the heater input to track the surrogate temperature, as shown in Figure 3c. Only the temperature model (which is more reliable than the grain growth model) is needed to enforce the input constraint.

10

(a)

(b)

(c)

Figure 3: Controller Structures: (a) Direct output feedback (b) MPC (c) Inner-outer loop control

Figure 4: Graph structure for consensus evaluation

To enforce grain size consensus between the different regions, we construct a directed graph G (Figure .4) to connect Nr nodes (zone grain sizes) and Ne directed edges, as in [19]. Let D be the Ne × Nr graph incidence matrix. Then grain size consensus is equivalent to Dz = 0. 3.2. Direct Output Feedback 3.2.1. Trajectory-based Control Given z ∗ (t) corresponds to the input u∗ (t) (within the input constraint), then z˙ − z˙ ∗ = g(F (u)) − g(F (u∗ )).

(6)

Let δu = u − u∗ , δz = z − z ∗ , and φ(δu) = g(F (δu + u∗ )) − g(F (u∗ )), we have we have δ z˙ = φ(δu),

(7)

Due to the fact that temperature, and grain growth, are collocated with the heater lines, higher current should lead to faster grain growth. This leads to the positive sector boundedness condition of φ(·): δuT φ(δu) ≥ αδuT δu 11

(8)

where α is a positive scalar. This is numerically verified by uniformly sampling 10,000 δu around the reference input u∗ . Around the reference input corresponding to 420o C base temperature, the lower bound is found numerically as α = 4.1667 × 10−6 µm2 · s−1 · mA−1 . It follows that a proportional output feedback δu = u − u∗ = −KP δz,

(9)

where KP is positive definite, is globally stabilizing, i.e., z(t) → z ∗ (t) as t → ∞. In fact, a large class of passive feedback control law is stabilizing. This is easily shown by using a Lyapunov function candidate [28] V =

1 T 2 δz KP δz.

The

derivative along (7) is 2 V˙ = δz T KP φ(−KP δz) ≤ α kKP δzk .

(10)

Since Kp > 0, this implies δz → 0 asymptotically (in fact, exponentially).

The reference input u∗ (t) may be solved from the specified z ∗ (t) via inverse

problem if the model is perfectly known. As this is almost never the case, we may use an integral term to estimate u∗ : u = −KP (z − z ∗ ) − KI

Z

(z − z ∗ ) dt,

(11)

where KI is also a positive definite matrix. Note that KI needs to be chosen sufficiently small to ensure stability. Integral control means infinite DC loop gain which also helps reject low frequency disturbances. Because of the heater input saturation, integral anti-windup should be applied. For enforcing both consensus and tracking, we design KP and KI as: K P = βP L + γ P K

(12a)

KI = βI L + γI K

(12b)

where βP , βI , γP , γI are positive tuning parameters. The feedback gain L is a weighted graph Laplacian corresponding to the consensus graph described at the end of Section 3.1: L = DT W D 12

(13)

The constant gain K may be any positive definite matrix and is simply chosen as the identity to simplify gain tuning. 3.2.2. Consensus-based Control Assume that a heater array input u∗ is chosen to establish a uniform base temperature y ∗ and the corresponding uniform grain growth trajectory z ∗ (i.e., grain sizes are in consensus). This implies Dz ∗ = 0, where D is the incidence matrix corresponding to a chosen consensus graph. Consider a consensus-based feedback using the weighted graph Laplacian L = DT W D: u = u∗ − βL Lz = u∗ − βL DT W Dz = u∗ − βL DT W Dδz.

(14)

where βL is a positive tuning parameter. This is exactly the same as trajectorybased control with Kp = βL L. From (10), we have kLδzk → 0, or Dz →

0, asymptotically, implying asymptotic consensus. Furthermore, as u → u∗ asymptotically, z converges to z˙ ∗ = g(F (u∗ )). 3.3. Model Predictive Control

Direct output feedback takes advantage of only the positivity in the combined heater array and grain growth model. It is robust with respect to the model information but does not consider performance such as rate of convergence and does not account for the input saturation constraints. In this section, we consider the model predictive control (MPC) method which uses the complete model information to optimize for the input over a finite time horizon subject to the constraints. To speed up computation, we consider the linearized model around the desired trajectory (u∗ , y ∗ , z ∗ ). The constrained finite horizon optimization then becomes quadratic programming (QP) problem which can be efficiently solved.

13

3.3.1. Trajectory-based Control In trajectory-based MPC, the objective function penalizes both the tracking error and the consensus error, as well as the input magnitude: J=

min δu(k+j|k), δz(k+j|k), j=0,...,NP −1

NX P −1 j=0

√ {w1 k W Dδz(k + j|k)k2 + w2 kδz(k + j|k)k2 (15) + w3 kδu(k + j|k)k2 }

subject to:

where

√

δz(k + j + 1|k) = δz(k + j|k) + Bk δu(k + j|k)

(16a)

δz(k|k) = δzm (k)

(16b)

umax ≥ δu(k + j|k) + u∗ ≥ 0.

(16c)

W is a diagonal weighting matrix, δu(k + j|k) is the sequence of predic-

tive control with NP as the planning horizon, δz(k + j|k) is the predicted output based on the linearized dynamics and predicted inputs δu(k + j|k), wi ’s are weighting coefficients, umax is the input constraint, Bk = Ts ∇u [g(F (u))]|u=uk is the linearized (simplified) FEM and MC model, δzm (k) is the measured output at k th time step, regarded as the initial condition of the discrete-time system in (16a), Ts is the sampling period. 3.3.2. Consensus-based Control For the consensus-only case, we only penalize the consensus error and the input norm: min δu(k+j|k), δz(k+j|k), j=0,...,NP −1

J=

NX P −1 j=0

√ {w1 k W Dδz(k + j|k)k2 + w2 kδu(k + j|k)k2 }.

under the same constraints as above, (16b)–(16c).

(17)

Both optimizations are

quadratic programming problems and may be readily solved using standard packages such as the cvx-MATLAB package [29].

14

3.4. Inner-outer Loop Control An important issue to consider for model-based controller is the model uncertainty. The heater array model can be effectively calibrated based on temperature measurements, but the grain growth dynamics for a specific sample is ¯ highly uncertain. The activation energy Q and the grain growth constant K may vary greatly with the deposition condition or the lifetime of the sample. To address that, we propose an inner-outer loop control structure, as illustrated in Figure 5. The idea is to use temperature as a surrogate input for grain growth control based on grain growth measurement, and then use heater input to track the required temperature. We shall see that the stability of the grain growth feedback loop does not require explicit model information. However, the bound on the heater input needs to be reflected as a constraint on the surrogate temperature input. This only requires the FEM model which is more accurate. A key step to show the robustness of the grain growth feedback loop is the sector bound of g in (4): δy T (g(y ∗ + δy) − g(y ∗ )) ≥ βδy T δy.

(18)

For specific y ∗ , we have numerically verified that β > 0. For example, for a base temperature of y ∗ = 420o C, the lower bound may be numerically computed as β = 8.6152 × 10−5 µm2 · s−1 · o C−1 . 3.4.1. Trajectory-based Control Similar to (11), consensus-augmented PI feedback may be used for the surrogate temperature yd to achieve trajectory-based control: Z I P yd = y ∗ − ζavg Kavg (z − z ∗ )dt − ζavg Kavg (z − z ∗ ) − ζL Lz

(19)

P I where ζavg , ζavg , ζL are positive scalar gains, Kavg is a positive definite matrix,

L is the same graph Laplacian matrix as in (13), y is the base temperature for the desired trajectory z ∗ , uniform for all zones. We use averaged PI feedback to compensate for overall grain size deviation from the desired trajectory combined with the consensus feedback to reduce spatial difference in grain growth rate. 15

Figure 5: Inner-outer loop control structure

The closed loop stability follows Because of the positive boundedness of g, the same procedure as in (10) may be applied to show the closed loop exponential stability if yd in (19) can be achieved. 3.4.2. Consensus-based Control For consensus-based control, feedback is only applied around the operating point (y ∗ , z ∗ ): yd = y ∗ − ζL Lz

(20)

Similar procedure as (10) may be applied to show the convergence of Dz → 0, implying consensus. 3.4.3. Heater array model inversion and adaptive gain adjustment To find u to achieve yd in (19) or (20) requires solving (5). The function F is monotone so the inverse function F −1 exists. Assume y ∗ = F (u∗ ) is achievI P able, then we can iteratively adjust the gains, (ζavg , ζavg , ζL ) in the trajectory-

based control and ζL in consensus-based control until a feasible F −1 (yd ) is achieved. In this paper, we use a limited-memory Broyden-Fletcher-GoldfarbShanno (BFGS) algorithm [30].

16

4. Results 4.1. Biased MC Simulation We consider 2D grain growth in a 1mm × 1mm thin film sample. The grain structure is initialized through uniform Voronoi tessellation with average grain diameter of 3 pixels (0.6µm). The temperature is set at 350o C for the left half domain and 450o C for the right half domain. Simulation stops at physical time t ≈ 100min. The final grain structure is shown in Figure 6, in which the grain size is around 0.8 µm in the left half domain and around 4.0µm in the right half domain. Grain size is measured using line-interception method [31] at various locations along the horizontal direction. This grain structure is used as the initial condition for all control simulations in this section.

Figure 6: A non-uniform grain structure is created using biased MC simulation (t ≈ 100min). The initial condition is a uniformly tessellated structure on a 5000×5000 lattice with grain diameter at approximately 3 pixels (0.6µm).

17

4.2. Grain Growth Control Three control methods described in Section 3 are applied to the simulated grain growth based on the full-order combined FEM and MC grain growth model for the heater array design in Figure. 1a (with ten heater lines, so Nr = 10). The controller tuning is conducted with the simplified model (4)–(5). For both the trajectory-driven control and consensus-driven control, a base temperature of 420o C is chosen for the reference grain growth trajectory. The substrate temperature is chosen as 390o C so that the heater array can provide temperature range from 390 ˜ 450o C. The constraints for heater array current is set as umax = 200mA. Control update time interval is approximately Ts ≈ 10 min. Assuming a 5000 × 5000 2-dimensional SEM image at 6µs dwell time for the grain size distribution measurement (based on the 5000×5000 grid used in the MC simulation), the image acquisition time is less than 3 minutes. The remaining time should be sufficient for the image analysis to extract the average grain size in each zone. Because for every MC step, the physical time increment varies with the current temperature field and local tmc , the k th control update is applied at every first MC steps when t > kTs . Closed-loop responses of the direct output feedback method are shown in Figure 7. In the trajectory-based control case shown in Figure. 7, the initial heater array input is 20mA in every heater line. Under the proposed PI feedback, consensus and tracking are both achieved at approximately 1200 min. The grain size profile also becomes flat at the end (t = 2000 min). Although grain uniformity is ultimately achieved, grain size in the right half domain undergoes a drop from 500 min to 1000 min. In the consensus-based case shown in Figure. 7, after the consensus is achieved around 700 min while the overall growth rate follows the base temperature. Figure 8 presents the final microstructure (at 2000 min) using direct output feedback method for the trajectory-based case. Compared to Figure. 6, the spatial grain size uniformity has been restored. Figure 9 shows the closed-loop response of MPC method. The sampling period Ts is chosen as 10min. Optimization is carried out every 2 time steps with the initial portion (first 2 steps) of the optimal input sequence applied to 18

140 120

z =D2 (µ m 2)

Grain Size D (µ m)

14

100

z1 z2 z3 z4 z5 z6 z7 z8 z9 z10 zr

80 60 40 20 0

t=0 t=100min t=500min t=1000min t=1500min t=2000min

16

0

500

1000 Time(min)

1500

12 10 8 6 4 2 0

2000

0

0.2

(a)

14 Grain Size D (µ m)

100 z1 z2 z3 z4 z5 z6 z7 z8 z9 z10 zr

80 60 40 20 0

500

1000 Time(min)

1

t=0 t=100min t=500min t=1000min t=1500min t=2000min

16

120

z =D2 (µ m 2)

0.8

(b)

140

0

0.4 0.6 X (mm)

1500

12 10 8 6 4 2 0

2000

0

(c)

0.2

0.4 0.6 X (mm)

0.8

1

(d)

Figure 7: Closed loop response of average grain size using direct output feedback method. (a), (b): Average grain size evolution of 10 heating zones and grain size profile evolution for the trajectory-based control. (c), (d): Average grain size evolution of 10 heating zones and grain size profile evolution for the consensus-based control.

19

Figure 8: Restored uniform grain structure at the end of the simulation, t = 2000 min, with the direct output feedback method applied to achieve trajectory-based control (corresponding to Figure 7). The final average grain size is D ≈ 11µm.

the combined FEM and MC simulation. Basic “warm-up” strategies are adopted between consecutive optimizations: part of the optimal solution from the last optimization (at time step k1 ) δuopt (k1 + j|k1 ), where j = 2, 3, ..., NP − 1 is used to initialize the current optimization (at time step k2 , where k1 + 2 = k2 ) δu(k2 + j|k2 ), where j = 0, 1, ..., NP − 3, and the last two time steps in the planning horizon are initialized as 0. The relative weighting w1 : w2 : w3 between consensus error, tracking error and input deviation in (15) is selected as 1000:1000:1, and that between consensus error and input deviation w1 : w2 in (17) selected as 1000:1. Even though a linear model is used to approximate the nonlinear function g(F (·)), the performance is still superior than direct output feedback: In the trajectory-based case shown in Figure. 9, consensus and tracking are both achieved at approximately 500 min. In the consensus-

20

based case shown in Figure. 9, it takes only around 300 min to achieve desired consensus. For the inner-outer loop controller, the approximated heater array model is used for solving the inverse problem in the outer loop to ensure the attainability of yd . Simulation results are shown in Figure 10 for both trajectory-based and consensus-based control. Although not performing as well as the linear MPC method, this approach still outperforms the output feedback method. Table 1 summarizes the performance of three methods under trajectorybased control in terms of tracking error and consensus error. Similarly, Table 2 compares the three methods towards a consensus-based control in terms of only consensus error. Performance of the same controller setting on the simplified model is also presented as a comparison. The direct output feedback method is easiest to implement with guaranteed stability but cannot achieve optimal performance due to strong coupling between adjacent zones and the input constraints. The other two approaches both perform better than the direct output feedback due to their model-based nature. The linear MPC method achieves the best performance amongst the three. The inner-outer loop method performs slightly worse than the MPC method, but is more robust to the expected large uncertainties in the grain growth process. It is interesting to note that, for trajectory-based control, the Inner Outer Loop approach achieves better performance than Direct Output Feedback not only in terms of the consensus error, but also the tracking error. We posit it’s because the overall MIMO system is nominally decoupled in the grain growth control loop, leading to a more organized formation as shown in the closed-loop response (Figure 10a).

5. Robustness of MPC MPC shows the best performance, but it also requires the most amount of model information. The direct output feedback is not dependent on any model, and inner-outer loop control only needs the thermal model. MPC may therefore be susceptible to modeling errors in the grain growth process, which

21

140 120

z =D2 (µ m 2)

Grain Size D (µ m)

14

100

z1 z2 z3 z4 z5 z6 z7 z8 z9 z10 zr

80 60 40 20 0

t=0 t=100min t=500min t=1000min t=1500min t=2000min

16

0

500

1000 Time(min)

1500

12 10 8 6 4 2 0

2000

0

0.2

(a)

14

100 80 60 40 20 500

1000 Time(min)

1500

Grain Size D (µ m)

z1 z2 z3 z4 z5 z6 z7 z8 z9 z10 zr

0

1

t=0 t=100min t=500min t=1000min t=1500min t=2000min

16

120

z =D2 (µ m 2)

0.8

(b)

140

0

0.4 0.6 X (mm)

12 10 8 6 4 2 0

2000

(c)

0

0.2

0.4 0.6 X (mm)

0.8

1

(d)

Figure 9: Closed loop response of average grain size using linear MPC method. (a), (b): Average grain size evolution of 10 heating zones and grain size profile evolution for the trajectorydriven objective. (c), (d): Average grain size evolution of 10 heating zones and grain size profile evolution for the consensus-driven objective.

22

140 120

z =D2 (µ m 2)

z1 z2 z3 z4 z5 z6 z7 z8 z9 z10 zr

80 60 40 20 0

500

1000 Time(min)

1500

Grain Size D (µ m)

14

100

0

t=0 t=100min t=500min t=1000min t=1500min t=2000min

16

12 10 8 6 4 2 0

2000

0

0.2

(a)

14 Grain Size D (µ m)

100 z1 z2 z3 z4 z5 z6 z7 z8 z9 z10 zr

80 60 40 20 0

200

400 600 Time(min)

1

t=0 t=100min t=500min t=1000min t=1500min t=2000min

16

120

z =D2 (µ m 2)

0.8

(b)

140

0

0.4 0.6 X (mm)

800

12 10 8 6 4 2 0

1000

(c)

0

0.2

0.4 0.6 X (mm)

0.8

1

(d)

Figure 10: Closed loop response of average grain size using inner-outer loop control structure as a trajectory-guided-consensus approach. (a), (b): Average grain size evolution of 10 heating zones and grain size profile evolution for the trajectory-driven objective. (c), (d): Average grain size evolution of 10 heating zones and grain size profile evolution for the consensus-driven objective.

23

Table 1: Simulation performance comparison of trajectory-driven grain growth based on three control methods. Performances are evaluated in terms of tracking and consensus error. k · k2 denotes L2 norm.

Direct Model

Metrics

Output

MPC

Feedback FEM + MC

Simplified

MPC

Inner-

Wrong

outer

Model

Loop

kz − z ∗ k2 √ k WDzk2

8.4×104

3.4×104

9.3×104

5.4×104

2.4×105

1.0×105

2.8×105

1.3×105

kz − z ∗ k2 √ k WDzk2

7.4×104

3.2×104

8.5×104

5.2×104

2.1×105

9.7×104

2.6×105

1.3×105

Table 2: Simulation performance comparison of consensus-driven grain growth based on three control methods. Performances are evaluated in terms of tracking and consensus error. k · k2 denotes L2 norm.

Direct Model

FEM + MC Simplified

Metrics √ k WDzk2 √ k WDzk2

Output

MPC

Feedback

MPC

Inner-

Wrong

outer

Model

Loop

1.6×105

8.6×104

2.3×105

1.3×105

1.2×105

8.3×104

1.4×105

1.2×105

may be caused by variation of the material property or even the thin film sample preparation process. To evaluate the robustness of MPC, we vary the grain ¯ in the control design while keeping the actual K ¯ growth model parameter K ¯ affects MPC in two in MC grain growth simulation the same. The error in K aspects: 1. The design of the reference input u∗ would be based on the wrong nominal model; 2. The linearized system used in the optimization contains an additional scaling in the input matrix, B = Ts ∇u [g(F (u))]|u=u∗ .

For trajectory-based control, the effect of incorrect reference input u∗ can be

mitigated by removing the penalty on the input deviation, w3 , in (15). However,

24

an incorrect linearized system (the matrix B) may still lead to undesired behav¯ 0 = 7.43 × 106 in iors of the controller, as shown in Figure 11. In this case, K ¯ applied for MC simulathe nominal model, 10 times smaller than the actual K

tion. Here, the model error actually increases the MPC gain, resulting in input saturation, especially during the initial control updates, which greatly increase the overall temperature and growth rate (see the grain size profile at 100 min in Figure 11). Convergence to the reference trajectory and consensus are still achieved, however, at a later time around 1300 min.

(a)

(b)

Figure 11: Closed-loop response of average grain size using linear MPC method corresponding ¯ 0 = 7.43 × 106 . Trajectory-based control to an incorrect nominal grain growth model, where K is applied here. The penalization on input deviation (w3 as in (15)) is set as 0. (a) Average grain size evolution of 10 heating zones. (b) Grain size profile evolution.

For consensus-based control, penalty on the input deviation should not be set to zero because a reference input u∗ is necessary to ensure the overall uniform growth after consensus has been achieved. Figure 12 shows the consensusbased closed-loop response using linear MPC method corresponding to the same incorrect nominal grain growth model as in Figure 11. The effects of both incorrect linearized system and incorrect reference input are evident in this case. Although consensus is first achieved around 300 min (average grain size around 6.4 µm), divergence is observed in the subsequent response, which is mainly due to the incorrect linear model around the reference input. In addition,

25

¯ 0 much lower than resulted from the incorrect reference input u∗ that assumes a K ¯ the overall grain growth ends up much faster than the reference the actual K, trajectory z ∗ . Performance of MPC with incorrect model is also included and compared with the other two control methods in Table 1 and 2. Overall, with an incorrect model, MPC is still able to track the reference trajectory but with degraded performance. For consensus-based control, MPC corresponding to an incorrect model can still achieve consensus but fails to maintain it well in the subsequent growth.

(a)

(b)

Figure 12: Closed-loop response of average grain size using linear MPC method corresponding ¯ 0 = 7.43 × 106 . Consensus-based control to an incorrect nominal grain growth model, where K is applied here. (a) Average grain size evolution of 10 heating zones. (b) Grain size profile evolution.

6. Conclusions Control of material microstructure and grain growth has long been experiencebased and open-loop in nature. We propose a closed-loop methodology to actively control the grain growth by adjusting distributed heat input based on average grain sizes in multiple zones. A distributed heater array is assumed to spatially control the temperature distribution inside a thin film sample and the grain growth. The biased Monte Carlo method is combined with FEM to

26

simulate the grain growth in response to distributed heating. For the control development, we use a simplified grain growth model from the parabolic grain growth rule. Three different control methods are developed to achieve either trajectory-based or purely consensus-based grain growth control: direct output feedback, linear MPC and an inner-outer loop approach. The controllers are tuned using the simplified model and then applied to the complete Monte Carlo and FEM simulation for comparison. All three methods successfully achieve the control objectives, but differ in terms convergence rates and the required levels of model information for their implementation.

Direct output feedback does

not require explicit model parameters but cannot achieve optimal performance due to the strong coupling between adjacent heating zones and the input saturation constraints. The MPC approach addresses these limits and achieves the best performance among three methods, but requires the full model information which made it susceptible to the expected large uncertainties in the grain growth model. The inner-outer loop method uses temperature as a surrogate input and closes the nominally decoupled grain growth control outer loop with model-free feedback. This approach takes advantage of the comparatively more accurate heater array model, but possesses better robustness compared to the MPC method as the grain growth model is not explicitly used. Future investigation includes in-situ experiments integrating automatic microstructure image processing and feedback grain growth control.

Acknowledgment The author would like to thank G.P.S. Balasubramanian for designing the micro-heater array, and R. Hull and G. Kane for providing FIB images and helpful discussions on experimental methods. This work is supported primarily by the National Science Foundation award CMMI-1334283 DMREF: Real Time Control of Grain Growth in Metals, and CMMI-1729336 DMREF: Adaptive Control of Microstructure from the Microscale to the Macroscale, and in part by the Center for Automation Technologies and Systems (CATS) under a block

27

grant from the New York State Empire State Development Division of Science, Technology and Innovation (NYSTAR).

Appendix A. Bias Monte Carlo Method for Grain Growth Simulation in a Non-uniform Temperature Field The biased Monte Carlo model for grain growth simulation has been presented in [21, 22]. This section provides a brief summary. The starting point is the classic Potts framework [32]. The spatial domain is discretized into a set of lattice sites, each assigned a grain orientation (usually represented as grain ID). The set of adjacent lattice sites with identical grain ID is defined as a grain. The evolution of grain structure is simulated by a random walk process driven by the objective of lowering the system’s interfacial free energy. In addition to this basic framework, we use the temperature field (from FEM) to bias the MC sampling to model abnormal grain growth [33]. In every iteration of the basic MC model (under uniform temperature), a lattice site A with crystallographic orientation SA is selected through uniform sampling, then the procedure shown in Figure. A.13 is executed. Assuming the number of total lattice points is N , then every Monte-Carlo step (δtmc = 1) contains N MC iterations. To link grain growth to the physical time, the following widely-accepted exponential growth rule is used [26, 27]: Q ¯ Dn − D0n = Kexp(− )t RT

(A.3)

where D denotes the mean grain size, D0 is the initial grain size, t the physical ¯ the grain growth constant, Q the activation energy, R the Boltzmann time, K constant, T the temperature (in Kelvin). The grain growth exponent, n, is Q generally accepted as 2 for copper [27]. The Arrhenius term exp(− RT ) is related

with grain boundary mobility, which increases with T . On the other hand, isotropic MC grain growth also follows: Dm − D0m = λm Ctmc 28

(A.4)

1: procedure REORIENT(A) 2:

if ∃B ∈ N (A) s.t.SA 6= SB then

. N (A) denotes A’s 8-neighbors

3:

Randomly choose a new SN from A ’s 8 neighbors

4:

Compute the free energy change due to reorientation: ∆E = J

X

(δSA SB − δSN SB );

(A.1)

B∈N (A)

. δab is the Kronecker-delta, J is the unit grain boundary energy Compute the reorient probability:   1 P =  exp(− ∆E )

5:

RTs

, ∆E ≤ 0

(A.2)

, ∆E > 0

. Ts denotes the simulation temperature [21].

Assign SA ← SN according to P

6: 7:

end if

8: end procedure Figure A.13: Pseudo-code for lattice site reorientation

Equating the average grain size D in Eqns (A.3) and (A.4) and raising to the physical grain growth exponent n yields [21] λn



D0 λ

m

+ Ctmc

n m

¯ = D0n + Ktexp(

−Q ) . RT

(A.5)

Taking a temporal differential increment, we can now relate a differential MC step dtmc to a differential physical time increment dt n  m m −1 n n D0 −Q ¯ λ + Ctmc C dtmc = Kexp( ) dt . m λ RT

(A.6)

Solving the above equation for the differential MC step dtmc , and separating the terms that are spatially dependent from those that are spatially independent, we obtain ¯ mK δtmc (X) = δt · ·h n {z } | nCλ X independent |

exp(− RTQ(X) ) n im −1
D0 m + Ct (X) mc λ {z }

29

X dependent

.

(A.7)

Hence, δtmc (X) ∝ h

exp(− RTQ(X) ) := H(X) n im −1
D0 m + Ct (X) mc λ

(A.8)

where X denotes any lattice site. Note that δtmc is equal to the expectation of times a lattice point being selected in one MC step, which is scaled through the introduction of Ps (X) := H(X)/H(X)max ,

(A.9)

known as the site selection probability, where H(X)max is the largest H(X) value throughout the entire lattice. At lattice sites with H(X)max , Ps (X) = 1, all selections are accepted so that δtmc = 1 for each MC step. In contrast, some selections will be rejected at lattice sites where Ps (X) < 1. Declaration of Competing Interests The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: References References [1] G. F. Vander Voort, S. R. Lampman, B. R. Sanders, G. J. Anton, C. Polakowski, J. Kinson, K. Muldoon, S. D. Henry, W. W. Scott Jr, ASM handbook, Metallography and Microstructures 9 (2004) 44073–0002. [2] A. Rollett, F. Humphreys, G. S. Rohrer, M. Hatherly, Recrystallization and related annealing phenomena, Elsevier, 2004. [3] C. Stubbington, A. Bowen, Improvements in the fatigue strength of Ti-6Al4V through microstructure control, Journal of Materials Science 9 (1974) 941–947. 30

[4] M. Jahazi, B. Egbali,

The influence of hot rolling parameters on the

microstructure and mechanical properties of an ultra-high strength steel, Journal of Materials Processing Technology 103 (2000) 276 – 279. [5] M. M. Kirka, Y. Lee, D. A. Greeley, A. Okello, M. J. Goin, M. T. Pearce, R. R. Dehoff, Strategy for texture management in metals additive manufacturing, JOM 69 (2017) 523–531. [6] M. A. Gama, M. Mahfouf, Model-based optimisation and control of the hot-rolling process for the design of steel microstructure, IFAC Proceedings Volumes 40 (2007) 255 – 260. 12th IFAC Symposium on Automation in Mining, Mineral and Metal Processing. [7] Y. Liu, M. S. Greene, W. Chen, D. A. Dikin, W. K. Liu, Computational microstructure characterization and reconstruction for stochastic multiscale material design, Computer-Aided Design 45 (2013) 65–76. [8] M. A. Gallivan, Optimization, estimation, and control for kinetic Monte Carlo simulations of thin film deposition, in: 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475), volume 4, pp. 3437–3442 vol.4. [9] X. Tang, B. Rupp, Y. Yang, T. D. Edwards, M. A. Grover, M. A. Bevan, Optimal feedback controlled assembly of perfect crystals, ACS Nano 10 (2016) 6791–6798. [10] Y. Lou, P. D. Christofides, Feedback control of surface roughness using stochastic PDEs, AIChE Journal 51 (2005) 345–352. [11] Z. K. Nagy, R. D. Braatz, Robust nonlinear model predictive control of batch processes, AIChE Journal 49 (2003) 1776–1786. [12] Z. K. Nagy, Model based robust control approach for batch crystallization product design, Computers & Chemical Engineering 33 (2009) 1685–1691.

31

[13] M. R. A. Bakar, Z. K. Nagy, C. D. Rielly, Seeded batch cooling crystallization with temperature cycling for the control of size uniformity and polymorphic purity of sulfathiazole crystals, Organic Process Research & Development 13 (2009) 1343–1356. [14] C.-A. Lin, Y.-K. Jan, Control system design for a rapid thermal processing system, IEEE Transactions on Control Systems Technology 9 (2001) 122– 129. [15] E. Dassau, B. Grosman, D. R. Lewin, Modeling and temperature control of rapid thermal processing, Computers & Chemical Engineering 30 (2006) 686–697. [16] M. H. Farshidianfar, A. Khajepour, A. Gerlich, Real-time control of microstructure in laser additive manufacturing, The International Journal of Advanced Manufacturing Technology 82 (2016) 1173–1186. [17] M. H. Farshidianfar, A. Khajepour, A. P. Gerlich, Effect of real-time cooling rate on microstructure in laser additive manufacturing, Journal of Materials Processing Technology 231 (2016) 468–478. [18] M. Murakami, M. Moriyama, S. Tsukimoto, K. Ito, Grain growth mechanism of Cu thin films, Materials Transactions 46 (2005) 1737–1740. [19] C. Zheng, G. P. S. Balasubramanian, Y. Tan, A. M. Maniatty, R. Hull, J. T. Wen, Simulation, microfabrication, and control of a microheater array, IEEE/ASME Transactions on Mechatronics 22 (2017) 1914–1919. [20] C. Zheng, Y. Tan, J. T. Wen, A. M. Maniatty, Finite element model based temperature consensus control for material microstructure, in: 2015 American Control Conference (ACC), pp. 619–624. [21] Y. Tan, A. M. Maniatty, C. Zheng, J. T. Wen, Monte Carlo grain growth modeling with local temperature gradients, Modelling and Simulation in Materials Science and Engineering 25 (2017) 065003.

32

[22] C. Zheng, Y. Tan, J. T. Wen, A. M. Maniatty, Material grain growth consensus control: A multi-zone heating approach applied on a MonteCarlo model, in: 2016 American Control Conference (ACC), pp. 3255– 3260. [23] R. Olfati-Saber, A. Fax, R. M. Murray, Consensus and cooperation in networked multi-agent systems, Proceedings of the IEEE 95 (2007) 215– 233. [24] S. A. Kalogirou, S. Panteliou, A. Dentsoras, Artificial neural networks used for the performance prediction of a thermosiphon solar water heater, Renewable Energy 18 (1999) 87 – 99. [25] B. Karlik, A. V. Olgac, Performance analysis of various activation functions in generalized MLP architectures of neural networks, International Journal of Artificial Intelligence and Expert Systems 1 (2011) 111–122. [26] J. Gao, R. Thompson, Real time-temperature models for Monte Carlo simulations of normal grain growth, Acta Materialia 44 (1996) 4565 – 4570. [27] A. Gangulee, Structure of electroplated and vapor-deposited copper films. III. recrystallization and grain growth, Journal of Applied Physics 45 (1974) 3749–3756. [28] H. K. Khalil, Nonlinear Systems, Pearson, 3 edition, 2001. [29] M. Grant, S. Boyd, Y. Ye, cvx: MATLAB software for disciplined convex programming, http://cvxr.com/cvx, 2008. [30] R. H. Byrd, P. Lu, J. Nocedal, C. Zhu, A limited memory algorithm for bound constrained optimization, SIAM Journal on Scientific Computing 16 (1995) 1190–1208. [31] R. Allen, Standard test methods for determining average grain size (F112), Annual Book of ASTM Standards, Metal-Mechanical Testing; Elevated and Low Temperature Tests; Metallography (1999). 33

[32] D. Z¨ ollner, P. Streitenberger,

Three-dimensional normal grain growth:

Monte Carlo Potts model simulation and analytical mean field theory, Scripta Materialia 54 (2006) 1697–1702. [33] B. Radhakrishnan, T. Zacharia, Monte Carlo simulation of grain boundary pinning in the weld heat-affected zone, Metallurgical and Materials Transactions A 26 (1995) 2123–2130.

34