Chilankamol Sunny

Satyajit Das
email: satyajitdas@iitpkd.ac.in

Kevin J M Martin
email: kevin.martin@univ-ubs.fr

Philippe Coussy
email: philippe.coussy@univ-ubs.fr

Hardware Based Loop Optimization for CGRA Architectures

Keywords: Coarse grained reconfigurable array (CGRA), Loop optimization, Hardware loop, CDFG transformation

With the increasing demand for high performance computing in application domains with stringent power budgets, coarse-grained reconfigurable array (CGRA) architectures have become a popular choice among researchers and manufacturers. Loops are the hot-spots of kernels running on CGRAs and hence several techniques have been devised to optimize the loop execution. However, works in this direction are predominantly software-based solutions. This paper addresses the optimization opportunities at a deeper level and introduces a hardware based loop control mechanism that can support arbitrarily nested loops up to four levels. Major contributions of this work are, a lightweight Hardware Loop Block (HLB) for CGRAs that eliminates control instruction overhead of loops and an acyclic graph transformation that removes loop branches from the application CDFG. When tested on a set of kernels chosen from various application domains, the design could achieve a maximum of 1.9× and an average of 1.5× speed-up against the conventional approach. The total number of instructions executed is reduced to half for almost all the kernels with an area and power consumption overhead of 2.6% and 0.8% respectively.

Introduction

Coarse Grained Reconfigurable Array (CGRA) architectures have proven to be good targets as specialized hardware in low power embedded applications, such as Wireless Sensor Networks (WSN), Internet of Things (IoT) and Cyber Physical Systems (CPS). Due to the regular structure with several highly optimized processing elements (PEs) and simple interconnects, CGRAs provide high performance and energy efficiency. Analyzing the energy consumption of the PEs, it is inferred that the scope of improvement lies in the control path rather than on individual units [START_REF] Vadivel | Loop overhead reduction techniques for coarse grained reconfigurable architectures[END_REF]. Since CGRAs are mostly used to accelerate the computeintensive portions of an application, specifically the loops, optimizing the control path for loops will result in significant improvement in energy efficiency.

Widely reported works in CGRA implementations [START_REF] Das | An energy-efficient integrated programmable array accelerator and compilation flow for near-sensor ultralow power processing[END_REF] [11] [START_REF] Prabhakar | Plasticine: A reconfigurable architecture for parallel patterns[END_REF] rely on software based loop optimizations to reduce the control flow bottleneck. Popular techniques like loop unrolling [START_REF] Vadivel | Loop overhead reduction techniques for coarse grained reconfigurable architectures[END_REF] [START_REF] Dragomir | Extending loop unrolling and shifting for reconfigurable architectures[END_REF], modulo scheduling [START_REF] Park | Edge-centric modulo scheduling for coarse-grained reconfigurable architectures[END_REF] [START_REF] Hamzeh | Epimap: Using epimorphism to map applications on cgras[END_REF] and polyhedral model based optimizations [START_REF] Liu | Polyhedral model based mapping optimization of loop nests for cgras[END_REF] are broadly used in CGRAs. All these approaches exploit the parallelism available in the innermost loop body and reduce the number of branch execution by transforming the loop to a data flow graph (DFG). However, with the growing complexity of the applications, the loop bounds are becoming larger and number of loop nests as well as total number of loops in an application are increasing. Reducing only the branch instruction in the innermost loop is not sufficient anymore to get high energy efficiency for a whole application. To deal with outer loops, additional instructions to perform iteration counter update, terminating condition checking and branching are required. These additional instructions diminish the overall energy efficiency. Therefore, any effort in the direction of eliminating these non-contributing loop control instructions will aid in optimizing the control path of the whole application and achieving better performance and energy gain.

In this paper, we introduce a novel hardware based loop optimization technique for CGRAs which eliminates the control flow bottleneck in loops. As the target CGRA, we choose the state-of-the-art Integrated Programmable Array (IPA) architecture [START_REF] Das | A heterogeneous cluster with reconfigurable accelerator for energy efficient near-sensor data analytics[END_REF]. In this CGRA, to optimize the loop control bottleneck, loop unrolling is used. On top of this we propose the hardware based loop optimization to enhance the overall performance and energy efficiency. As of now, the optimization is limited to loops with statically known bounds. Experimental results show that the proposed solution reduces the total number of instructions executed to half of that in the baseline software based implementation and achieves a maximum of 1.9× performance gain for a wide range of image and signal processing applications. The first major contribution of this work is a novel lightweight Hardware Loop Block (HLB) for CGRAs which eliminates branch instruction overhead for loop increments or decrements. The second major contribution is in the compilation flow. We propose an acyclic graph transformation in the compilation to transform the software loops in the application CDFG eliminating loop branches. Experiments show that the total number of branch instruction execution is eliminated up to an average of 30× compared to that of the software based approach. Synthesis results show that the added hardware in the baseline architecture brings an area and a power consumption overhead of 2.6% and 0.8% respectively.

The rest of the paper is organized as follows. Section 2 presents the related works on loop optimizations with hardware support, from both general purpose processor and CGRA architectural domains. Section 3 introduces the baseline architecture and loop model and explains the motivation behind this work. Section 4 is dedicated to detail the proposed approach and present the implementation. In Section 5 experimental results are discussed and section 6 concludes the paper.

Related Works

Hardware based loop optimizations primarily are of two types, zero overhead looping extensions and instruction memory hierarchy optimizations [START_REF] Vadivel | Loop overhead reduction techniques for coarse grained reconfigurable architectures[END_REF]. Zero overhead looping refers to automatically updating the iteration count and taking branch decisions by using dedicated hardware units. One major optimization on instruction memory hierarchy favoring loops is to include loop buffers. Loop buffers cache the instructions that make up the loop body to reduce the instruction fetch cost. We have seen hardware loop support on processors from the early x86 processor, the ISA of which included loop instruction and rep instruction prefix. The rep prefix triggered repeated execution of a single instruction. Many early ISAs included rep-like prefixes to support single cycle loops. Program address based and instruction count based zero overhead loop accelerations were proposed for DSP [START_REF] Tsao | Hardware nested looping of parameterized and embedded dsp core[END_REF] and RISC [START_REF] Kavvadias | Elimination of overhead operations in complex loop structures for embedded microprocessors[END_REF] architectures. They could even provide multi-level loop support by using stack or scratch pad memory. In VLIW architectures [START_REF] Mathew | A loop accelerator for low power embedded vliw processors[END_REF], a special hardware unit is attached with every issue slot to automatically generate the instruction address. By employing loop buffers, DSP [START_REF] Bajwa | Instruction buffering to reduce power in processors for signal processing[END_REF] and x86 processors could significantly reduce their energy consumption. The parallel ultra-low power (PULP) cluster architecture [START_REF] Gautschi | Near-threshold risc-v core with dsp extensions for scalable iot endpoint devices[END_REF] is a multi-core platform with hardware support for loops. PULP implemented hardware loop by including an additional controller and a set of registers to store the loop information to its RISC-V cores. By having two sets of dedicated registers, the design could support two level nesting of loops. The instruction pre-fetch-buffer of the architecture acts as a loop cache if the loop body would fit into it, amplifying the effect of the hardware loop.

State-of-the-art CGRA architectures are adopting to implement hardware based loop solutions. Vadivel et al. [START_REF] Vadivel | Loop overhead reduction techniques for coarse grained reconfigurable architectures[END_REF] proposed a zero-overhead loop accelerator (ZOLA) for CGRA architectures. It supports nesting of loops up to four levels. ZOLA is enabled using a custom instruction after configuring all the loop parameters in the configuration register. The configuration of inner loops are kept intact until the outermost loop finishes execution. This prevents the reuse of configuration registers within a nested loop. Hence the total number of loops in a nested structure is limited by the number of configuration registers. For instance, in a ZOLA implementation employing four configuration registers, a nested loop of depth (number of levels) four can have only one loop at each level. In this paper, we propose a hardware based loop implementation which can support up to four levels of nesting and arbitrary number of sibling loops. The term sibling loops refers to loops at the same level or having the same loop as immediate parent. Furthermore, loop configuration is done inline with the execution eliminating the need for an extra configuration phase. ZOLA tries to reduce energy consumption by freeing the extra hardware units its reference architecture employs to implement loop control flow. However, the replacement of units is possible only when they are not used for any other computation. The solution we propose aims to minimize the number of loop control instructions executed by the baseline model, rather than trying to reduce the PE usage.

LASER [START_REF] Balasubramanian | Laser: A hardware/software approach to accelerate complicated loops on cgras[END_REF] is another attempt to take hardware support for optimizing the loop execution. It is a hardware-software hybrid solution designed particularly for loops with imperfect nesting and random nesting of conditionals. Nesting of loops is called perfect if all the assignment instructions are inside the innermost loop; otherwise it is called imperfect nesting. In LASER, the compiler flattens nested loops into a single loop with conditional statements and fuses the truepath and false-path operation nodes. The instruction fetch unit (IFU) is modified to select and execute either the true-path or the false-path instruction based on the branch outcome. By enhancing IFU, LASER tries to better support loops

Background and Motivation

This section gives an introduction to IPA [START_REF] Das | An energy-efficient integrated programmable array accelerator and compilation flow for near-sensor ultralow power processing[END_REF], the baseline architecture used to implement our design. The software based loop model of IPA as well as the problem statement is discussed in this section.

Architecture

The IPA integrated system consists of an array of inter-connected homogeneous processing elements (PEs), a global configuration memory (GCM), and a DMA controller. It is loosely coupled with a host CPU. A multi-banked tightly coupled data memory (TCDM) facilitates data communication between IPA and the CPU. Fig. 1 gives the IPA integrated system architecture. Each PE in the PE array houses a functional unit (FU), a controller, a decoder, an instruction register file (IRF), a regular register file (RRF), a constant register file (CRF) and router connections at the input and output. Global configuration memory stores the configurations (instructions and constants) for all the PEs in the PE array, which will be loaded to IRFs and CRFs of individual PEs, prior to execution. At each clock cycle, the PE controller fetches an instruction from IRF and FU executes it to completion.

Baseline Loop Model

IPA manages loop control flow by supporting branch instructions. The innermost loop is partially unrolled to exploit instruction and data level parallelism. However, once the loop body completes an iteration, the iteration counter is updated, an unconditional jump to the instruction which checks the terminating condition is performed, the condition check is done and then a conditional jump is performed either to the loop start or to the subsequent instruction in the kernel. Evidently, the counter update, condition checking and branching instructions which contribute only to the loop control mechanism are executed as many times as does the loop body. Performance bottleneck imposed by this overhead can be promptly inferred by comparing the number of loop control instructions against the number of kernel instructions executed. For an imperfectly-nested loop structure of depth d, total number of kernel instructions executed (N ke ) can be represented as:

N ke = d i=1 n j=1 p * i k=1 q (1)
Total number of loop control instructions executed (N lc ) can be computed as:

N lc = q * d i=1 n j=1 i k=1 q (2)
where n is the number of loops at level i, p is the number of kernel instructions in j th loop at level i, q is the loop count of loop at level k and q is the number of loop control instructions executed per iteration. Loop count is the number of times the loop is to be executed. In the proposed design, we target to eliminate the loop control instruction to improve performance and energy efficiency.

Proposed Architecture and Compilation Flow

We propose a novel hardware based approach to implement loop control structure on CGRAs. Features which distinguish this design from other solutions are:

i Support for arbitrarily nested loops: the proposed solution can handle loop nesting up to four levels, which suffices for the studied benchmarks. The loops can be perfectly or imperfectly nested. It can handle arbitrary number of sibling loops. ii Synchronous termination of loops at multiple levels: if an instruction happens to be the last instruction of multiple loops in the nested structure, the iteration count of each of those loops are updated in a single cycle in which the innermost loop terminates. If it was the last iteration of those loops, then they all will be terminated instantly. iii On the go loop configuration: loop configuration is done inline with the execution. A separate pass through the loop structure is not required to set the configuration registers. Hardware loop units are incorporated into each PE in the PE array, modifying its architecture as shown in Fig. 2. In the individual PEs, HLB is integrated with the Controller, redefining its branch unit. The branch unit is responsible for updating the program counter (PC), which points to the next instruction to be fetched from the IRF. Loop characteristics such as start address, end address and iteration count are used to configure the loop and need to be stored in registers.

To support four levels of nested loop, HLB reserves four configuration registers, one for each level and an extra register for loopLevel which holds the current level number. Each time a new loop is encountered, loopLevel is incremented by one, signifying a move one level deeper and when a loop runs to completion, loopLevel is decremented by one indicating a move up to the parent loop. At each level, the corresponding loop configuration register gets active. This register selection is controlled by the Loop Select Control Unit (LSCU). For instance, when a loop at level 2 is in execution, LSCU chooses the configuration register, R2 (Fig. 2). For every loop that appears at a particular level, the same register is reused to hold the configuration data facilitating the support for arbitrary number of sibling loops at each level. At each clock cycle, the design compares the current PC value to the configured loop end instruction address to detect the branch-point. Until PC reaches the loop end, execution happens sequentially incrementing PC by one. When the PC and end address values evaluate to be equal, the loop count value is decremented by one and the PC is set to loop start address as long as the loop count is a non-zero value. Loop count becoming zero implies that the current loop has run to its completion and hence it is set as a trigger to decrement loopLevel. Consequently, the parent loop parameters become the current loop configuration, if there is any. PC is updated as PC + 1 which takes the control to its parent loop or to the rest of the kernel as the case may be. If the loop at hand was the outermost loop, then loopLevel will be zero indicating that no loop is in execution.

When a child loop finishes its execution, the design compares PC with the end address of parent loop as well. This is to support synchronous configuration update and/or termination of loops sharing same end address. If the PC value and the parent end address are found to be the same, the loop count of the parent loop is decremented. If the parent loop was in its last iteration, then the loopLevel decrement will be by two. If not, loopLevel will be decremented by one and PC will be updated with the loop start address of the parent loop. This checking repeats for the subsequent levels as well causing loopLevel to be decremented by one, two, three or four depending on the scenario. LSCU facilitates this conditional update on PC and loopLevel. This feature allows the required termination of loops and/or loop count update at multiple levels in a single cycle, benefit of which is particularly evident in the case of perfectly nested loops. It is evident that current PC value and the loop end/start addresses are involved in choosing the next instruction. Since these addresses may vary across PEs, a completely centralized implementation of HLB is not possible. The baseline compilation flow and the required modifications are discussed in the next section.

Compilation flow

IPA supports block oriented execution of applications. It implies that the kernel to be accelerated is partitioned into single-entry-single-exit blocks of instructions called basic blocks (BBs) and only one basic block is executed at a time. Therefore the configuration as well as the assembly code are prepared in a block by block fashion. The IPA compiler takes as input the CGRA model and the high level application expressed as a control and data flow graph (CDFG) in which nodes represent BBs and edges represent the control flow between them. Each BB itself is represented as a data flow graph (DFG), BB = (D, O, A) where D is the set of data nodes, O is the set of operation/activity nodes and A is the set of arcs representing data dependencies. PE array (PEA) of the IPA is modelled by a bipartite directed graph of operator nodes and register nodes. CGRA model is the time extended model of PEA in which the timing is represented by connections between registers and operators. It is homomorphic to the basic block DFG and hence the mapping of a DFG onto the PEA is equivalent to finding a DFG in the PEA graph [START_REF] Das | Efficient mapping of cdfg onto coarse-grained reconfigurable array architectures[END_REF].

The baseline compilation flow (F ig. 3) starts with generating a CDFG from the application expressed in C language. This is done using GCC 4.8. Next step is to select the BB to be mapped, for which depth first search (DFS) strategy is used. The first BB is mapped without any constraints. However, to map the subsequent BBs, the constraints imposed by the maps of previously mapped BBs are to be considered. For this reason, the first map out of the many mappings generated for the previous BB is chosen as the start point. Constraints from this mapping are inferred and consolidated and this happens in the update constraints phase. To schedule the DFG of each BB, a backward traversal list scheduling algorithm is used. Operation nodes are listed by a priority order based on mobility and fanout. Once the highest priority node is identified, the compiler finds a placement for it in the PEA model and schedules it. If a placement solution could not be found, then the DFG is transformed to improve the placement possibilities. Graph transformations increase the physical or temporal distance between source and sink to keep data dependencies or reduce the node fan outs by operation splitting. The modified DFG then goes into the scheduling and placement phase. If the transformation does not make any change in the DFG, then that calls for a backtracking. Another map from the mapping list is chosen, constraints are updated and the process starts all over again. If the mapping is successful, a stochastic pruning is applied on the partial mapping set to prevent it from growing exponentially. Once the assembly file containing the map for all the BBs is prepared, it is fed to the assembler together with the ISA. Assembler then generates the configuration for the PE array.

In the proposed model, a CDFG transformation phase is introduced as the first step in the compilation flow. Since the transformation is done prior to mapping, it is termed pre-mapping CDFG transformation. Additionally, the priority order in which operations are listed is redefined so that loop activity node would be scheduled to fire as the first activity in the concerned BB. The IPA ISA is extended with two dedicated instructions and the assembler is also modified accordingly. The newly introduced step, CDFG transformation and the modified blocks like Scheduling & Placement, Assembler and IPA ISA are highlighted in Fig. 3. The pre-mapping CDFG transformation is detailed next.

We introduce two new terms, Hardware Loop Header (HLH) and Hardware Loop Terminal (HLT) to title the entry and exit BBs of the loop. HLH is the BB which dominates all other blocks which constitute the loop body and HLT is last block in that set. Algorithm 1 lists down the steps in transforming the input CDFG. The transformation is applied to loops of depth up to four as the hardware unit is so designed. Iteration count of the loop is computed from the initial and final values of loop control variable and the step by which the variable gets updated in each iteration of the loop. The loop step can be greater than or equal to one and the update can be increment or decrement. The algorithm identifies the first BB in the set of BBs that forms the loop body, which is essentially the true path successor of loop condition checking BB. This BB is designated as the HLH. The last BB in the loop body referred as loop latch in the baseline model is set as the HLT. A loop activity node and a data node holding the loop count value are inserted into the HLH. Address of the last instruction in HLT is the loop end address. As discussed earlier, loop characteristics such as start address, end address and loop count are used to configure the hardware loop and that explains the significance of HLH and HLT. As the next step, the BB that does the condition checking which was the loop header in the baseline model and the back jump from HLT to this BB are eliminated from the CDFG. This essentially does a cyclic to acyclic graph transformation on that part of the CDFG which represents the loop. If the loop control variable is not used as an operand in any of the BBs, then the nodes that correspond to its initialization and increment/decrement operations are removed. If the BB housing those nodes does not include any other node, then that BB as a whole will be pulled out of the CDFG. If the loop control variable is used inside the loop body, then those operation nodes will be preserved and the variable gets updated by the the step value, each time the loop iterates. Once this transformation is done for each loop in the CDFG, the algorithm does another pass through the list of loops to merge the HLTs of outer and inner loops if possible. The merging is performed recursively down the levels as long as it is feasible to do so. By doing this merge, these loops will have the same end address which would trigger the hardware unit to update the loop count of all those loops together as the inner one finishes its last iteration. As the algorithm runs to completion transforming the CDFG, we are left with a very simple acyclic graph representing a significantly reduced set of instructions. Fig. 4 shows a sample program, the corresponding CDFG and the transformed version of it.

Experiments and Results

Experimental Setup

In this section, we analyse the implementation results providing performance and area comparison for a set of commonly used DSP applications. To ensure an unbiased evaluation and test the range of applicability, we have chosen a set of loop intensive kernels from various application domains with varying structure The IPA achieves an energy efficiency improvement up to 18×, with an average of 9.23× (as well as a speed-up up to 20.3×, with an average of 9.7×) [START_REF] Das | An energy-efficient integrated programmable array accelerator and compilation flow for near-sensor ultralow power processing[END_REF] compared to a RISC-V core [START_REF] Gautschi | Near-threshold risc-v core with dsp extensions for scalable iot endpoint devices[END_REF] specialized for ultra-low power nearsensor processing.

Implementation results

This section describes the implementation results for the baseline IPA architecture and the proposed design. Both the designs were synthesized with Cadence Genus Synthesis Solution 17.22-s017 1 using 90nm CMOS technology libraries.

The IPA implementation on which the experiments are conducted is configured to have a 4×2 PE array with each PE housing a 21×64-bits IRF, a 20×32-bits CRF and a 32×8-bits RRF. On top of this baseline IPA, Hardware Loop Block (HLB) module was integrated to support the proposed optimization. Fig. 5 (a) and (b) reports the area and power consumption overhead of the proposed HLB of 2.6% and 0.8% respectively over the baseline architecture.

Performance analysis

Table 2 presents the comparison of executed instructions in the software based baseline and the proposed hardware based optimization. It is observed that the number of instructions executed is reduced to half for almost all the kernels; achieving an average gain of 2×. The obtained results clearly emphasize that by eliminating loop control instructions, the total number of instructions executed can be greatly reduced. The same is illustrated in Fig. 6 which gives a comparison of the number of instructions executed by accelerating a simple loop on the baseline and proposed models for a 2×1 CGRA. The loop control instructions that would be eliminated by the proposed technique are highlighted in red (Fig. 6(b)). Since the loop variable i is used as an operand in the loop body, its initialization and update operations are preserved. By eliminating the highlighted instructions, total number of instructions executed is reduced from 46 to 18 achieving a gain of 2.56× (Fig. 6(e)). It is interesting to note that the gain on reducing the number of branches is 12× ((Fig. 6(d)) which is multiple times of what obtained for the total number of instructions. At any given instant, every PE in the baseline PE array will be executing instructions from the same basic block (BB). To achieve this block-level synchronicity, branch instructions (JMP/CJMP) that govern the control flow between BBs are copied to every PE. Therefore the effect of eliminating one branch instruction from the code will be reflected in the execution profile by many folds. On top of that, more than 50% of the loop control instructions that get eliminated would be branch instructions as can be seen from Fig. 6(b). This explains the huge difference on the gain obtained for branches and total number of instructions. Speed-up or the performance gain achieved by the enhancement over the baseline model is computed as the ratio of execution cycles. As depicted in Table 3, an average gain of 1.5× is recorded for the set of kernels we considered. Table 4 gives the comparison of the number of branches executed. As expected, the proposed solution outperforms the baseline by a huge margin. An average gain of 30× and a maximum of 78× is recorded. To have the instruction execution on PEs synchronized to the BB boarders, each PE in the PEA would be executing the branch instruction simultaneously. Therefore, eliminating one branch instruction will reduce the number of cycles by one but the reduction on the number of branches will be by the number of PEs. This is the reason why we do not see a direct correlation between the reduction in branches and execution cycles. The total number of basic blocks is also found to be reduced (Table 5) in the proposed solution for all the kernels, effect of which is already seen in the number of branches. It is by the pre-mapping CDFG transformation that the BBs and the branch operations got eliminated. Therefore, the gain in terms of number of BBs and branches acts as an efficiency measure of the CDFG transformation as well. In the next phase, we will explore energy efficiency of the proposed approach.

Total No. of Instructions Executed i=0 JMP i<10 CJMP A[i]=A[i]+1 A[i+1]=A[i+1]+1 i=i+2 JMP i<10 CJMP A[i]=A[i]+1 A[i+1]=A[i+1]+1 i=i+2 JMP i<10 CJMP ... Gain Achieved 2.56x Baseline Model Proposed Model for(i=0;i<10;i=i+2) { A[i]=A[i]+1; A[i+1]=A[i+1]+1; } i=0 i<10 A[i]=A[i]+1 A[i+1]=A[i+1]+1 i=i+2

Conclusion

The performance bottleneck imposed by the overhead of having loop control instructions in kernel execution is often substantial. It is impossible to get rid of all the control bottlenecks with software only loop optimizations. In this paper, we introduced a hardware based loop control mechanism for CGRAs targeted to reduce the number of non-contributing control operations. By moving the loop control to hardware, the proposed model could achieve an average speedup of 1.5× and reduce the number instructions executed to half, against the baseline CGRA architecture running software based loops. Up to 78× and an average of 30× gain is observed on reducing the number of executed branch instructions with an area and power consumption overhead of 2.6% and 0.8% respectively. The significant improvement noted on a set of kernels with varying loop structure suggests that the discussed design can be presented as a generic loop optimization technique to a wide range of applications.

Fig. 1 :

 1 Fig. 1: IPA Integrated System with conditional blocks rather than optimizing the general execution of loops. The basic implementation of looping mechanism remains to be software based. It does not eliminate the condition checking or branch instructions. The solution we propose eliminates those loop control instructions.

Fig. 2 :

 2 Fig. 2: Baseline PE Architecture extended with HLB 4.1 Hardware Loop Block (HLB)

Fig. 3 :

 3 Fig. 3: Baseline Compilation Flow extended with CDFG Transformation

Fig. 4 :

 4 Fig. 4: (a) Sample program; (b) Corresponding CDFG; (c) Transformed CDFG

Fig. 5 :

 5 Fig. 5: (a) Area breakdown in PE; (b) Power consumption breakdown in PE

Fig. 6 :

 6 Fig. 6: (a) Partially unrolled kernel and corresponding CDFG; (b) Instructions executed on the baseline model, and (c) proposed model for a 2×1 CGRA; (d) Total number of branches executed on baseline vs proposed models; (e) Total number of instructions executed on baseline vs proposed models

Table 1 :

 1 Maximum depth and iteration counts of loops for the listed kernels

	Kernel			Depth of Nesting Max. # of Loop Iterations
	Floyd Warshal	3		60×60×60 = 216,000
	Jacobi-1D		2		20×(28+28) = 1,120
	Seidel-2D		3		20×38×38 = 28,880
	FIR filter		2			190×10 = 1,900
	2D Convolution	4		80×60×3×3 = 43,200
	Erosion			4		78×58×3×3 = 40,716
	Dilation			4		78×58×3×3 = 40,716
	Matrix Multiplication	3		32×32×32 = 32,768
	Matrix Addition	2			32×32 = 1,024
	9.5% DIV_SQRT REG FILE 5.3%	HLB 2.6%	OTHERS 3.5%	38.9% CRF	RT, DIV_SQ 9.0% REG FILE,	HLB, 0.8%	OTHERS, 3.7%	CRF,
					20.1%		50.1%
	ALU
	9.1%		INST MEM 31.1%		ALU, 6.5%	9.9% INST MEM
			(a)				(b)

Table 2 :

 2 Comparison of the total number of instructions executed between the baseline and the proposed models

	Kernel	Baseline Model Proposed Model Gain Achieved
	Floyd Warshal	3,514,585	2,204,373	1.59×
	Jacobi-1D	36,525	13,895	2.63×
	Seidel-2D	1,627,047	827,389	1.97×
	FIR filter	25,858	11,795	2.19×
	2D Convolution	1,066,898	1,009,438	1.06×
	Erosion	67,379	48,722	1.38×
	Dilation	92,379	37,154	2.49×
	Matrix Multiplication	410,130	176,399	2.33×
	Matrix Addition	12,818	5,391	2.38×
	Average			2.00×

Table 3 :

 3 Comparison of latency in terms of execution cycles between the baseline and the proposed models

	Kernel	Baseline Model Proposed Model Gain Achieved
	Floyd Warshal	2,203,210	1,548,434	1.42×
	Jacobi-1D	15,197	11,268	1.35×
	Seidel-2D	670,945	582,652	1.97×
	FIR filter	15,327	9,358	1.64×
	2D Convolution	538,623	399,280	1.35×
	Erosion	41,726	28,059	1.49×
	Dilation	41,750	28,013	1.49×
	Matrix Multiplication	271,930	140,713	1.93×
	Matrix Addition	9,777	6,569	1.49×
	Average			1.50×

Table 4 :

 4 Comparison of the number of branches executed between the baseline and the proposed models

	Kernel	Baseline Model Proposed Model Gain Achieved
	Floyd Warshal	1,325,288	446,770	2.97×
	Jacobi-1D	11,815	164	72.04×
	Seidel-2D	487,064	6,248	77.96×
	FIR filter	9,126	764	11.95×
	2D Convolution	432,646	274,084	1.58×
	Erosion	41,661	14,200	2.93×
	Dilation	41,661	9,370	4.45×
	Matrix Multiplication	139,526	4,164	33.51×
	Matrix Addition	4,358	69	63.16×
	Average			30.10×

Table 5 :

 5 Comparison of the number of basic blocks mapped between the baseline and the proposed compilation flow

	Kernel	Baseline Model Proposed Model Gain Achieved
	Floyd Warshal	16	13	1.23×
	Jacobi-1D	14	10	1.40×
	Seidel-2D	14	10	1.40×
	FIR filter	11	8	1.34×
	2D Convolution	24	20	1.20×
	Erosion	22	18	1.22×
	Dilation	24	20	1.20×
	Matrix Multiplication	19	14	1.56×
	Matrix Addition	11	7	1.57×
	Average			1.40×

of loop nesting. Finite Impulse Response (FIR) filter is included to represent the family of digital filters. By virtue of the four-level nesting of loops, the image processing kernels, 2D convolution, erosion and dilation will aid in a corner case testing of the proposed approach. The kernels, Jacobi-1D and Seidel-2D performing stencil computations are used in several different application domains including data mining and machine learning. Similar is the case with Floyd Warshal kernel which is used for solving all pair shortest path problems. Jacobi-1D performs computations with 3-point stencil pattern over one dimensional data whereas Seidel-2D is a computation over 2 dimensional data with 9-point stencil pattern providing a deeper loop to test with. The generic kernels, matrix multiplication and addition are also included. Altogether the set has a good mix of perfectly and imperfectly nested loops. A detailed characteristics of these kernels is listed in Table 1.

Performance of the hardware loop implementation is compared against that of the baseline IPA architecture that employs only software based loop opti-


Rhone P Brocha Silalahi

Tzu-Hao Chiu

Jhen-Heng Kao

Chun-Yen Wu

Chi-Wei Yin

Yu-Chiao Liu

Yuan Jang Chen

Jean-Yves Saillard

Ming-Hsi Chiang

C W Liu

Yu- Chiao Liu

Synthesis and Luminescence Properties of Two-Electron Bimetallic Cu-Ag and Cu-Au Nanoclusters via Copper Hydride Precursors

Keywords: superatom, bimetallic, luminescence, copper, silver, gold

published or not. The documents may come   L'archive ouverte pluridisciplinaire

ABSTRACT:

The synthesis, structural characteristics, and photophysical properties of luminescent Cu-rich bimetallic superatomic clusters [Au@Cu12(S2CN n Pr2)6(C≡CPh)4] + (1a + ), [Au@Cu12{S2P(OR)2}6(C≡CPh)4] + (2 + ), (2a

+ = i Pr; 2b + = n Pr),
[Au@Cu12{S2P(C2H4Ph)2}6(C≡CPh)4] + (2c + ), and [Ag@Cu12{S2P(O n Pr)2}6(C≡CPh)4] + (3 + ) were studied. Compositionally uniform clusters 1 + -3 + were isolated from the reaction of dithiolatostabilized, polyhydrido copper clusters with phenylacetylene in the presence of heterometal salts.

By using X-ray diffraction, the structures of 1a + , 2a + , 2b + , and 3 + were able to be determined.

ESI-mass spectrometry and elemental analysis confirmed their compositions and purity. The structural characteristics of these clusters are similar to display gold (or silver)-centered Cu12 cuboctahedra surrounded by six dithiocarbamate/dithiophosph(in)ate and four alkynyl ligands.

The doping of Au and Ag atoms into the polyhydrido copper nanoclusters significantly enhances their PL quantum yields from Ag@Cu12 (0.58%) to Au@Cu12 (55%) at ambient temperature in solution. In addition, the electrochemical properties of the new alloys were investigated by cyclic voltammetry.

Introduction

Bimetallic nanoclusters (NCs) are extensively studied because of their ability to tune the various properties (optical, catalytic, etc.) and possibly improve the stability of their monometallic counterparts in a controlled way. [START_REF] Bootharaju | Templated Atom Precise Galvanic Synthesis and Structure Elucidation of a [Ag24Au(SR)18] -Nanocluster[END_REF][START_REF] Barrabes | Racemization of chical Pd2Au36(SC2H4Ph)24: doping increases the flexibility of the cluster surface[END_REF] Photoluminescence (PL) is one of the major properties broadly explored in bimetallic NCs, [START_REF] Wang | Intensely luminescent gold(I)silver(I) cluster complexes with tunable structural features[END_REF] owing to its potential applications in bioimaging, cell labeling, chemical sensing, drug delivery, or phototherapy. 4 The PL of NCs can be tuned by several methods, such as aggregation-induced emission, [START_REF] Kang | Bimetallic Au2Cu6 Nanocluster: Strong Luminescence Induced by the Aggregation of Copper(I) with Gold(0) Species[END_REF] cluster-based networks via selfassembly of the NC building blocks, [START_REF] Huang | Hypersensitive dual-function luminescence switching of a silverchalcogenolate cluster-based metal-organic framework[END_REF] alloying metal kernels, [START_REF] Dhayal | Ag21{S2P(O i Pr)2}12] + : an eight-electron superatom[END_REF] and engineering peripheral ligands. [START_REF] Kang | Modulating photoluminescence of Au2Cu6 nanocluster via ligand-engineering[END_REF] However, bimetallic copper-based clusters with a high PL quantum yield (QY) are still rarely reported. Notable examples are Au2Cu6(PPh2Py)2(SC10H15)6 (QY=11.7%), 5a,8a [Au7Cu12(dppy)6(TBBT)6Br4](SbF6)3 (QY=13.2%), [START_REF] Song | The Structure of an Au7Cu12 Bimetal Nanocluster and Its Strong Emission[END_REF] and [AuCu14(SPh t Bu)12(PPh(C2H4CN)2)6] + (QY=71.3%). [START_REF] Song | Ultrabright Au@Cu14 nanoclusters: 71.3% phosphorecescence quantum yield in nondegassed solution at room temperature[END_REF] Hence, new synthetic methodologies for the isolation of copper alloy NCs with excellent quantum efficiency are still needed. In this research, we report new bimetallic twoelectron NCs based on copper that can be synthesized by both alloying the metal kernel and engineering the peripheral ligands to exhibit a high QY of 55%.

In previous studies, a series of copper-rich bimetallic 2-electron NCs, [M@Cu12(S2CN n Bu2)6(C ≡ CPh)4] + (M = Au, Ag), were obtained by a galvanic exchange strategy involving the reaction of [Cu13(S2CN n Bu2)6(alkynyl)4] + as a template with M(I) salts. 11 The doped species show differences in their optical and electronic properties and stability compared to their template precursor. In particular, the Au-doped cluster exhibits a high PL intensity with QY=59% at 77 K in MeTHF. 11 Later, we uncovered that similar bimetallic Cu-rich NCs stabilized by 1,1-dithiophosphate ligands can shift both emission and absorption spectra to higher energy and display high QY at ambient temperature.

Herein, we report the synthesis of Cu-Au and Cu-Ag clusters generated from the treatment of excess amounts of phenylacetylene with dithiolato-stabilized copper hydrides in the presence of Au(I) and Ag(I) salts. This methodology is totally different from our previously reported procedure of templated galvanic replacement. 11 These clusters have a centeredcuboctahedral M@Cu12 kernel surrounded by six dithiocarbamate/dithiophosph(in)ate and four alkynyl ligands with one counteranion to balance the cluster charge. These cationic NCs show idealized Td symmetry and possess two free ns valence electrons (ns = 13-6-4-1), making them pseudospherical 2-electron superatoms with a 1S 2 configuration. [START_REF] Walter | A unified view of ligand-protected gold clusters as superatom complexes[END_REF] Varying both the nature of M ACCEPTED MANUSCRIPT 3 (Ag, Au) and of the dithiolate ligands allows the tuning of the stability, optical and PL properties of the considered NCs, as described below. We previously reported the synthesis of 2-electron M@Cu12 (M = Ag, Au) clusters obtained from a galvanic replacement reaction using [Cu13{S2CN n Bu2}6(C≡CPh)4] + as a template. 11 In this paper, we describe a new synthetic procedure from as-synthesized copper(I)

Results and Discussion

Synthesis and Characterization

hydrides with terminal alkynes in the presence of Au(I) or Ag(I) salts (Scheme 1). This approach allows the use of either dithiocarbamate or dithiophosph(in)ate as stabilizing ligands.

An equimolar amount of Au(PPh3)Cl was added to a THF suspension of [Cu28H15{S2CN n Pr2}12](PF6), [START_REF] Edwards | Chinese Puzzle Molecule: A 15 Hydride, 28 Copper Atom Nanoball[END_REF] followed by the addition of fifteen equiv. of phenylacetylene. The mixture was stirred for 48 hours at 30°C. Purifications led to the generation of the (CuCl2) -salt of S8). In the negative-ion mode ESI-MS, a unique counteranion band is observed in the case of 2a + , corresponding to (CuCl2) -(m/z 134.9; calcd 134.8; Figure S9).

The 1 H and [START_REF] Edwards | Chinese Puzzle Molecule: A 15 Hydride, 28 Copper Atom Nanoball[END_REF] C NMR spectra of 2a S16-S18). These results are consistent with the fact that all the dithiolate and alkynyl ligands bind similarly to the Au@Cu12 kernel, as in 1a + .

The FT-IR spectra of clusters 2a + , 2b + , and 2c + exhibit v(PhC≡C -) stretching frequencies at 2014.3 (2a + ), 2018.9 (2b + ) and 1968.0 (2c + ) cm -1 (Figure S19-S21), which are lower than the value of the free phenylacetylene ligand (vC≡C = 2110 cm -1 ).

A similar synthetic procedure was conducted to obtain compound 3(PF6). An equimolar Clusters 1a + and 2a + were isolated as salts of [CuCl2] -with P21/c and P21/n space groups, respectively. 2b + and 3 + crystallized with Cl -(space group P(-)1) and PF6 -(space group P21/n) counteranions, respectively. The X-ray structures of 1a + , 2a + , 2b + , and 3 + are similar to that of [Au@Cu12{S2CN n Bu2}6(C≡CPh)4] + (1b + ), 11 as exemplified by the structural drawings of 1a + , 2b + , and 3 + , as shown in Figure 1. They all exhibit an M-centered Cu12 cuboctahedron, the six square faces of which are capped by the six dithiolato ligands in a (µ2, µ2) binding mode and its four triangular faces by the four alkynyl ligands in a (µ3-η 1 ) binding mode. Selected metric data of 1a

X-ray Photoelectron Spectroscopy

The oxidation state of the central metal in the cluster was evaluated by X-ray photoelectron spectroscopy (XPS). The XPS spectrum exhibits an Au 4f7/2 peak for clusters 1a A similar trend is found in the Ag 3d5/2 binding energy of cluster 3 + (367.8 eV, see Figure S31), which is intermediate between that of Ag metal (368.2 eV) and that of Ag(I) (367.5 eV).

The XPS features of all other elements were found exactly as expected (Figure S32-S36).

Optical Properties

The UV-vis absorption spectra of compounds 1a(CuCl2), 2a(CuCl2), 2b(Cl), 2c(CuCl2), and 3(PF6) are shown in Figure 2. Their corresponding λmax values are listed in or Ag), it is possible to evaluate their effect on the absorption features. It appears that the effect is negligible on both the low-and high-energy absorption bands. This result is somewhat surprising because DFT calculations on related models found a significantly larger HOMO-LUMO gap for gold-centered species. 11 A reasonable explanation is that in the case of 3, the low-energy band is associated with a HOMO → LUMO transition that is mixed with another transition of higher energy. 2). Such long emission lifetimes strongly suggest the emissive origin to be from a spinforbidden triplet excited state. The emission profiles of 1 + ~ 3 + are shown in Figures 3 andS42.

The photophysical data of their respective salts recorded at 298 K are summarized in Table 2 together with those of (1b)(CuCl2) 11 for comparison. Previously, we reported that 1b + emits intense red light with an emission wavelength centered at λem = 616 nm (λexc = 473 nm) at 77 K in solution, where the quantum yield achieves 59%. 11 For the emission λmax values, the room temperature λem values reported in Table 2 for clusters 1 + -3 + in solution can also be divided into 4 groups, namely, (1a In the solid-state (Figure S42), the room temperature emission wavelengths of 2a + , 2b + and 2c + (623, 638 and 617, respectively) barely change compared to their solution counterparts.

In the cases of 1a + and 1b + , it is redshifted to 724 and 765 nm, respectively. This behavior could be tentatively explained by the availability in 1a + and 1b + of the π*(dithiocarbamate) orbitals.

These MOs, which are situated not far above the three superatomic 1P LUMOs, 11 could, by simple C=N stretching, stabilize a low-lying triplet state that would be radiatively emitting only in the solid-state.

Ranging from 23% (1a + ) to 55% (2b + ), the QY reported in Table 2 is substantial for all gold-centered species. The much lower value obtained for 3 is consistent with the expected small spin-orbit coupling in the case of silver, as opposed to the heavier gold element, for which it favors spin intersystem crossing. [START_REF] Steinfeld | Molecules and Radiation, an introduction, modern molecular spectroscopy[END_REF] This result is also consistent with the fact that the Au-centered species exhibit significantly shorter PL lifetimes than the Ag-centered cluster. Going down to Cu renders spin-orbit coupling even more negligible, and consistently, [Cu@Cu12{S2CN n Bu}2(C≡ CPh)4](CuCl2) displays a very weak emission in the NIR region in a 2-MeTHF solution at 77 K and no emission at all at ambient temperature. 11 The results indicate that the PLQY for the twoelectron bimetallic alloy clusters follows the order Au@Cu12> Ag@Cu12 >> Cu@Cu12. Thus, gold doping is an efficient method for boosting the luminescent QY of bimetallic Cu-rich alloys, Whereas the solutions of dithiocarbamate species 1a + and 1b + have the largest radiative rate constant, they also have the largest nonradiative rate constant (together with 2c + ). With its lowest nonradiative rate constant, 2b + reaches the highest QY. Among the four gold-centered species, 2b + is the only one having chloride as a couteranion, and the others have a complex anion (CuCl2) -. An explanation (among others) for the limited nonradiative processes in 2b + is that the specific interactions with the chloride couteranion in the excited state could limit the nonradiative decay by enhancing the rigidity. More experiments with various salts of various gold/dithiophosphate relatives are needed to obtain a better understanding of the PL efficiency.

Electrochemical Investigations

The electrochemical behavior of clusters 1 + -3 The Au-centered clusters exhibit higher stability in the reduced state relative to the Agdoped derivative. Regarding the ligand influence, the dithiocarbamate-ligated clusters display greater susceptibility toward decomposition upon redox events. This observation is consistent with the results of time-dependent electronic spectra, revealing faster decay of the dithiocarbamate clusters compared with the corresponding dithiophosph(in)ate derivatives. The resulting electrochemical HOMO-LUMO gaps (E E g) are all approximately 2 eV (Table 3), and that of Ag derivative 3 is definitely lower than that of its Au congeners. These values are consistent with the optical HOMO-LUMO gaps computed from the lowest energy absorption band λmax values (Eg O ) as well as those computed from the recorded PL λm values (Eg PL ). The only inconsistent value is that of Eg O for 3 + , for which we suspect that the λmax transition is not associated with a pure HOMO→LUMO transition (see above). Indeed, DFT-computed HOMO-LUMO gaps on related species support the view of lower HOMO-LUMO gaps in the case of Agcentered species. 11 Within the Au-centered series, the increasing HOMO-LUMO gaps in the order 2c + > (2a + , 2b + ) > (1a + , 1b + ) are fully consistent with the variation of the PL λem wavelength, as well as the lowest optical λmax value. spectrophotometer. Emission spectra and lifetimes in solution were obtained in 2 mm cylindrical quartz cells in a spectroscopic quartz Dewar as described in elsewhere. [START_REF] Tsai | Are Very Small Emission Quantum Yields Characteristic of Pure Metal-to-Ligand Charge-Transfer Excited States of Ruthenium(II)-(Acceptor Ligand) Chromophores?[END_REF][START_REF] Tsai | Experimental and DFT Characterization of Metal-to-Ligand Charge-Transfer Excited States of (Rutheniumammine)(Monodentate Aromatic Ligand) Chromophores[END_REF] Emission spectra at room temperature were obtained by using a Xe lamp for wavelengths and an Oriel model 63358 or 63966 Quartz Tungsten Halogen (QTH) lamp for intensities. The emission spectra were collected using a HORIBA JOBIN YVON iHR 550 spectrometer with three gratings (300 l/mm, 600 nm blaze; 300 l/mm, 1 μm blaze; and 600 l/mm, 1 μm blaze) and a HORIBA Symphony

InGaAs-1700 (for the NIR) detector head was mounted on the exit port. This system was operated through the SynerJY software. The potentials are reported against the ferrocenium/ferrocene (Fc + /Fc) couple.

Synthesis of 1a(CuCl2):

In a flame-dried Shlenck tube, [Cu28H15{S2CN n Pr2}12](PF6), (0.1 g; 0.025 mmol) was suspended in THF (5 mL) along with Au(PPh3)Cl (0.012 g; 0.025 mmol) and phenylacetylene (40 uL; 0.375 mmol), the resulting mixture was stirred at 30°C for 48 hours. The solvent was evaporated under vacuum and residue washed with diethyl ether (3x15 mL) to remove impurities from the ligands. The red residue was extracted in methanol and passed through Al2O3. Finally, the solvent was evaporated to dryness under vacuum to get a pure dark red powdered 1a(CuCl2) in 203.26, 131.98, 128.28, 123.79, 77.81, 77.30, 59.53, 20.45, 11.19; FT-IR data in CsI pellet (cm -1 ); 2962. 4, 2932.5, 2873.3, 2018.7, 1592.3, 1304.9, 1238.8, 940.4, 893.8 128.27,123.79,75.80,74.83,53.84,23.26 ppm;31 P NMR (121.49 MHz,CDCl3); 101.73 ppm;

FT-IR data in CsI pellet (cm -1 ); 2979. 7, 2932.8, 2872.3, 2014.3, 1592.3, 1261.6 951.7, 887.0. Anal. Calcd for C68H104AuCl2Cu13O12P6S12: C, 29.40; H, 3.77; O, 6.91; S, 13.85. Found: C, 29.27; H, 3.57; O, 6.75; S, 13.70.

Synthesis of 2b(Cl):

In a flame-dried Shlenck tube, [Cu30H18{S2P(O n Pr)2}12], (0.05 g; 0.01 mmol) was suspended in THF (5 mL) along with Au(PPh3)Cl (0.005 g; 0.01 mmol) and phenylacetylene (25 µL; 0.18 mmol), the resulting mixture was stirred at 30°C for 48 hours. The solvent was evaporated under vacuum and residue washed with diethyl ether (3x15 mL) to remove impurities from the ligands.

The red residue extracted in methanol and passed through Al2O3. Finally, the solvent was remove impurities from the ligands. The black residue was extracted in methanol and passed through Al2O3. Finally, the solvent was evaporated to dryness under vacuum to get a pure black powdered 3(PF6) in 0.019 g (29.27 % based on Cu) yields. In this reaction, corresponding alkenes and [Cu8H{S2P(O n Pr)2}6] + have been produced as by products. 132. 17, 128.48, 122.76, 77.24, 72.06, 71.41, 23.11, 10.06 2, 2937.7, 2880.5, 2015.0, 1592.7, 1199.4, 913.1, 844.4. Anal. Calcd for C68H104AgCu12F6O12P7S12: C, 30.26; H, 3.88. Found: C, 29.86; H, 3.84.

X-ray Crystallography

Single crystals suitable for X-ray diffraction analysis of salts of 1a + , 2a + , 2b + , and 3 +

were obtained by slowly diffusing hexane into a concentrated acetone solution at -5 ºC temperature. The single crystals were mounted on the tip of glass fiber coated in paratone oil, then frozen at 150 K. Data were collected on a Bruker APEX II CCD diffractometer using graphite monochromated Mo Kα radiation (λ = 0.71073 Å). Data reduction was performed with SAINT, [START_REF]Software for CCDC Detector System; Bruker Analystic X-ray Systems[END_REF] Absorption corrections for the area detector were performed by using the SADABS program. [START_REF] Sheldrick | [END_REF] The structure was solved by direct methods and refined by least-squares against F2 using the SHELXL-97 package, 24 incorporated in SHELXTL/PC V5. 10.6. 25 Selected Crystallographic data is given in Table S1.

Associated Content

  amount of [Ag(CH3CN)4](PF6) was added to the THF suspension of copper(I) hydride [Cu30H18{S2P(O n Pr)2}12][START_REF] Barik | Polyhydrido Nanocluster with a Hollow Icosahedral Core: [Cu30H18{E2P(OR)2}12] (E= S or Se; R= nPr, iPr, or iBu)[END_REF] along with eighteen equiv. of phenylacetylene. The reaction was stirred for 48 hours at 30 °C. Purification of reaction products led to the isolation of [AgCu12{S2P(O n Pr)2}6(C≡CPh)4] + (3 + ) as a dark magenta precipitate in 29% yields. A band for[3 + ] at m/z 2554.30 (calcd 2554.30), consistent with its simulated isotopic pattern, is identified in the positive-ion ESI mass spectrum of compound 3 (Figures S22). The 31 P NMR signal of 3 + was observed at 105.71 ppm and that of its counterion PF6 -at -142.67 ppm, with a septet pattern (FigureS23). Similar to clusters[1 + ] and [2 + ], only one type of alkyl ligand is found in the 1 H and[START_REF] Edwards | Chinese Puzzle Molecule: A 15 Hydride, 28 Copper Atom Nanoball[END_REF] C NMR spectra (FigureS24-S25). The FT-IR spectrum exhibits v(PhC≡C -) stretching frequencies at 2015.0 cm -1 (FigureS26), similar to that reported for [Ag@Cu12{S2CN n Bu2}6(C≡CPh)4] + (2015 cm -1 ). 11 Because in the above clusters, the 13 metal atoms have a +11 global oxidation state (2electron superatoms), the reactions in Scheme 1 correspond to a partial reduction of the heterometal. The Au(I) or Ag(I) salts are reduced by the hydrides of the starting copper(I) hydrides, as evidenced by the production of H2, which can be detected by 1 H NMR spectroscopy during the reaction process. This route has been demonstrated in the fabrication of the 2-electron copper superatom, [Cu13H2(dtc)6(alkynyl)4] +, via the degradation of an intermediate cluster, [Cu15H2(dtc)6(alkynyl)6] + .[START_REF] Chakrahari | Isolation and Structural Elucidation of 15-Nuclear Copper Dihydride Cluster: An Intermediate in the Formation of a Two-Electron Copper Superatom[END_REF]

Figure 1 .

 1 Figure 1. (a) Single crystal X-ray structures of cluster cations 1a + , 2b + , and 3 + with hydrogen atoms omitted for clarity; (b) Thermal ellipsoid drawings (50% probability) of centered cuboctahedral kernels M@Cu12 (M = Au, Ag).

Figure 2 .

 2 Figure 2. UV-vis absorption spectra of salts of 1a + , 2a + , 2b + , 2c + , and 3 + in CH2Cl2. The inset shows photographs in CH2Cl2 at ambient temperature.

Figure 3 .

 3 Figure 3. Emission spectra of 1a + , 1b + , 2a + , 2b + , 2c + , and 3 + in MeTHF at ambient temperature.

Figure 4 .

 4 Figure 4. Cyclic voltammogram (black) and differential pulse voltammograms (blue and red line) of 2a + (0.6 mM) at 233 K in CH2Cl2 containing 0.1 M [ n Bu4N][B(C6F5)4] solution under N2 atmosphere.

  The detector heads was cooled to -90 o C and the spectrometers was purged with dry N2. The room temperature emission lifetimes were determined using a LTB Model MNL 103-PD nitrogen laser-pumped, a LTB Model DUL 100 Dye laser system for excitation, a Hamamatsu P928 PMT/E717-63 socket assembly mounted on a Jobin-Yvon H-100 spectrometer for detection in the visible region and a Hamamatsu NIR-PMT Modul H10330A-75 for collection in the NIR region. Both detector systems for detection in the visible and NIR regions were output digitized using the LeCroy WaveRunner 6030A software. Electrochemical measurements were recorded on a CH Instruments 733B electrochemical potentiostat using a gastight three-electrode cell under N2 at 273 K or 233 K. A vitreous carbon electrode (1 mm or 3 mm in diameter) and a platinum wire were used as working and auxiliary electrode, respectively. Reference electrode was a non-aqueous Ag + /Ag electrode (0.01 M AgNO3/0.1 M [nBu4N][BArF24], BArF24 -= B(3,5-C6H3(CF3)2)4 -). All experiments were measured in CH2Cl2 solutions with 1 mM analyte and 0.1 M [nBu4N][BArF24] supporting electrolyte concentrations and were recorded at a 100 mV/s scan rate unless otherwise stated.

3 (

 3 PF6): ESI-MS: m/z 2554.30 (M + ). 1 H NMR (300 MHz, d6-acetone): 7.46-7.67 (20H, C6H5), 4.07 (24H, OCH2), 1.40 (24H, CH2), 0.74 (36H, CH3) ppm; 13 C NMR (100.61 MHz, CDCl3):

+ , 2b + and 2c + show similar NMR patterns with one type of

	alkyl dithiophosph(in)ate and alkynyl ligand signal, indicating highly symmetric structures in
	solution (Figures S10-S15). This is also supported by their 31 P NMR spectra: 2a + at 101.73 ppm,
	2b + at 107.09 ppm, and 2c + at 81.09 ppm (Figure

Table 1 .

 1 This trend results from the smaller dithiocarbamate "bite distance" compared to that of a dithiophosph(in)ate. The Cu-S distances are slightly shorter in the case of the dithiocarbamate ligands. The Cu-C distances increase slightly in the order 1 > 2 > 3. Overall, the five structures are quite similar. Selected bond lengths (Å) for 1a + , 1b + , 2a + , 2b + , and 3 + .

		Mcent-Cu	Cu-Cu	Cu-S	Cu-C	C≡C
	1a +	2.7143(11)-2.7533(11)	2.5018(13)-2.9546(13)	2.265(2)-2.332(2)	2.011(8)-2.103(7)	1.199(11)-1.234(10)
		avg. 2.7356(11)	avg. 2.7352(13)	avg. 2.290(2)	avg. 2.046(8)	avg. 1.215(11)
	1b + 11	2.7102(7)-2.7735(7)	2.5283(9)-2.9134(9)	2.2638(14)-2.3208(15)	1.971(5)-2.089(5)	1.208(8)-1.218(7)
		avg. 2.7336(7)	avg. 2.7337(9)	avg. 2.2927(15)	avg. 2.037(6)	avg. 1.213(8)
	2a +	2.7575(11)-2.7942(10)	2.5499(14)-3.0291(14)	2.296(2)-2.334(3)	2.005(9)-2.081(9)	1.192(12)-1.218(12)
		avg. 2.7733(11)	avg. 2.7723(14)	avg. 2.313(3)	avg. 2.030(9)	avg. 1.208(12)
	2b +	2.7380(11)-2.7866(11)	2.5730(14)-2.9248(15)	2.283(3)-2.340(3)	2.003(10)-2.079(10)	1.197(13)-1.211(12)
		avg. 2.7641(11)	avg. 2.7631(15)	avg. 2.312(3)	avg. 2.031(10)	avg. 1.205(13)
	3 +	2.7394(6)-2.7916(7)	2.5416(8)-2.9022(8)	2.2938(13)-2.3325(12)	1.965(5)-2.080(4)	1.208(6)-1.220(6)
		avg. 2.7636(7)	avg. 2.7629(8)	avg. 2.3178(13)	avg. 2.018(5)	avg. 1.214(6)

+ , 2a + , 2b + , and 3 + are given in Table 1, together with that of 1b +11 for comparison. Not considering their alkyl and phenyl substituents, all the complexes have idealized Td symmetry. Their averaged Mcent-Cu and Cut-Cu distances are almost equal, although the latter can be divided into two groups. The dithiocarbamate derivatives 1a + and 1b + have metal-metal distances (~2.74 Å) slightly shorter than their dithiopho(phin)ate counterparts 2a + , 2b + , and 3 + (2.76-2.77 ACCEPTED MANUSCRIPT 7 Å).

Table 2 ,

 2 together with that of (1b)(CuCl2)11 for comparison. They exhibit remarkably similar optical features.Those of 1a + and 1b + (gold/dithocarbamates) are very close, as are those of 2a + and 2b + (gold/dithiophophates). Those of 2c (gold/dithiophosphate) are closer to those of 2a + and 2b + . A similar tendency can be noticed for 3 (silver/dithiophophates). Thus, a significant hypsochromic shift in the lowest energy band when going from dithiocabamate to dithiophosph(in)ate ligands can be traced. Since the low-energy band is a HOMO (1S)→LUMO (1P) transition, 11 it follows that this gap is slightly larger in the case of dithiophosph(in)ate ligands than in the case of dithiocarbamates. This is in line with the fact that in the case of the dithiocarbamate series, the 1P LUMOs are somewhat stabilized by π*(CN) levels.

11

With respect to the high-energy band, an opposite (bathochromic) shift is observed upon going from dithiocabamate to dithiophosph(in)ate ligands. Since 2b + and 3 + differ only by the nature of their heterometal (Au

Table 2 .

 2

	Compound	λmax(nm) [ε(M -2a m -1 )]	λem (nm)	Lifetime, τ(µs) One-exp a	Φem b	kobs, 1/τ(µs -1 ); one-exp	kRAD c (µs -1 )	kNRD d (µs -1 )
	1a(CuCl2)	311, 475, 508, 560 [8150]	657	0.76	0.23	1.3	0.30	1.0
		310, 476, 509,
	1b(CuCl2) 11	561	658	1.41	0.43	0.710	0.31	0.40
		[6400]
	2a(CuCl2)	326, 433 480,540 [8510]	637	3.20	0.32	0.31	0.10	0.21
	(2b(Cl)	324, 427, 480, 536 [5310]	642	3.67	0.55	0.27	0.15	0.12
	2c(CuCl2)	333, 427, 477, 533 [8340]	618	1.86	0.25	0.53	0.13	0.40
	3(PF6)	321, 476, 503, 537 [11700]	762	14.7	0.058	0.068	0.0039 0.064

298 K absorption, emission, and lifetime of 1a + , 1b + , 2a + , 2b + , 2c + , and 3 + in 2-MeTHF a mean emission lifetime (τ). b emission quantum yield (see page S1(SI)). c Φem= kRAD/kobs, kRAD = radiative rate constant and kobs = mean excited-state decay rate constant. d kobs = kRAD + kNRD, kNRD = nonradiative rate constant. e 77 K emission in 2-MeTHF glasses, weak emission intensity.

  Epc waves in the same region and two Epa oxidations, in which the second process is quasi-reversible. Dithiophosphinate species 2c + exhibits one Epc wave and three Epa processes, of which the second is quasi-reversible. Regarding 3 + , the Ag analog of 2b + , two irreversible reductions and four oxidations, of which the second is quasi-reversible, are recorded. From previous DFT calculations on related simplified models, 11 the reduction should correspond to electron transfer into the triply degenerate 1P LUMO, and the first two oneelectron oxides should concern the 1S HOMO. The mono-and di-oxidized species were computed to be thermally stable. According to the results of the electrochemical study, the dioxidized species is more stable than the mono-oxidized species.

	+ was investigated in CH2Cl2 solution at 233
	K. The results, shown in Figures 4, S49 and S50, are summarized in Table 3. Cyclic
	voltammograms (CVs) and differential pulse voltammograms (DPVs) of dithocarbamate species
	1a + and 1b + display one irreversible reduction process (Epc) at approximately -2.15 V and four

quasi-reversible oxidation processes (Epa) between -0.21 and +0.57 V. In the case of dithiophosphate derivatives 2a + and 2b + , the reversibility of the redox processes is improved, ACCEPTED MANUSCRIPT 13 with two quasi-reversible

Table 3 .

 3 Summary of redox potentials (V vs Fc + /Fc) and estimated band gaps (eV) of compounds 1-3. Potentials were determined by DPV. The measurements were collected at 233 K. All redox processes are irreversible unless they are labeled as ‡, indicating quasi-reversible processes. § Eg E and Eg O are the electrochemical and optical HOMO-LUMO gaps, respectively. Eg E = Epc-Epa, and Eg O = 1240/λmax, where λmax (in mm) corresponds to the lowest electronic transition. Eg PL = 1240/λem where λem (in mm) is the PL wavelength. ‡ Quasi-reversible. The silver-centered cluster appears to behave somewhat differently from its gold-centered relatives: whereas the latter exhibits good PL efficiency, the former only weakly emits. The largest recorded QY (55%) was obtained for one of the gold-centered dithiophosphate derivatives, which are the most thermally stable among the investigated families. Further research is planned to design new salts of [Au@Cu12(dithiolate)6(C≡CPh)4] + cationic clusters to enhance their QY and tune their emission wavelengths for various potential applications.

	Cluster Epa †	Epc †	Eg E §	Eg O §	Eg PL §
	1a +	-0.12, +0.17,	-2.14	2.02	2.21	1.89
		+0.37, +0.57
	1b +, 11	-0.21, +0.12,	-2.15	1.94	2.21	1.89
		+0.25, +0.54
	2a +	+0.26, +0.56 ‡	-1.96 ‡ ,	2.22	2.30	1.95
			-2.19 ‡
	2b +	+0.16, +0.65 ‡	-1.95 ‡ ,	2.11	2.31	1.93
			-2.17 ‡
	2c +	+0.19, +0.58 ‡ ,	-2.04	2.23	2.33	2.01
		+0.82
	3 +	-0.07, +0.15 ‡ ,	-1.95,	1.88	2.31	1.63
		+0.55, +0.76	-2.16
	†

  0.052 g (38.35 % based on Cu) yields. In this reaction, corresponding alkenes and [Cu8H{S2CN n Pr2}6] + has been produced as by products.

	1a(CuCl2): ESI-MS: m/z 2420.60 (M + ), 134.90 (M -). 1 H NMR (300 MHz, CDCl3): 7.26-7.49
	(20H, C6H5), 3.82 (24H, NCH2), 1.70 (24H, CH2), 0.86 (36H, CH3) ppm; 13 C NMR (100.61
	MHz, CDCl3):

  . Anal. , the resulting mixture was stirred at 30°C for 48 hours. The solvent was evaporated under vacuum and residue washed with diethyl ether (3x15 mL) to remove impurities from the ligands.The red residue was extracted in methanol and passed through Al2O3. Finally, the solvent was evaporated to dryness under vacuum to get a pure red powdered 2a(CuCl2) in 0.058 g (43.44 % based on Cu) yields. In this reaction, corresponding alkenes and [Cu8H{S2P(O i Pr)2}6] + has been produced as by products. ESI-MS: m/z 2642.10 (M + ), 134.90 (M -).[START_REF] Bootharaju | Templated Atom Precise Galvanic Synthesis and Structure Elucidation of a [Ag24Au(SR)18] -Nanocluster[END_REF] H NMR (300 MHz, CDCl3): 7.60-7.61 (20H, C6H5), 4.82 (12H, OCH), 1.17 (72H, CH3) ppm;13 C NMR (100.61 MHz, CDCl3): 132.82,

	Calcd for C74H104AuCl2Cu13N6S12: C, 34.77; H, 4.10; N, 3.29; S, 15.05. Found: C, 33.21; H, 4.22;
	N, 3.92; S, 14.68
	Synthesis of 2a(CuCl2):
	In a flame-dried Shlenck tube, [Cu20H11{S2P(O i Pr)2}9] (0.1 g; 0.031 mmol) was suspended in
	THF (5 mL) along with Au(PPh3)Cl (0.015 g; 0.031 mmol) and phenylacetylene (40 uL; 0.341
	mmol)

2a(CuCl2):

  ppm; 31 P NMR (121.49 MHz, d6acetone); 105.70, -143.0 (JP-F= 708.29 Hz) ppm; FT-IR data in CsI pellet (cm -1 ); 2969.

evaporated to dryness under vacuum to get a pure red powdered 2b(Cl) in 0.049 g (41.56 % based on Cu) yields. In this reaction, corresponding alkenes and [Cu8H{S2P(O n Pr)2}6] + have been produced as by products. Acknowledgment This work was supported by the Ministry of Science and Technology in Taiwan (MOST 109-2113-M-259-008) and the France-Taiwan ANR-MOST 2018 program (project Nanoalloys).

ACCEPTED MANUSCRIPT 24 25. SHELXL v6.14 (PC version) Program Library for Structure Solution and Molecular Graphics, Bruker Analytical X-ray Systems, Madison, WI, 1998.

For Table of Content only:

Compared to its homometallic Cu@Cu12 counterpart, the presence of an Au atom in the center of the cubocathedral (Au@Cu12) nanocluster significantly enhances its PL quantum yield to 55% at ambient temperature in solution.


Rahul N Chacko

François P Landes

Giulio Biroli

Olivier Dauchot

Andrea J Liu

David R Reichman

Elastoplasticity Mediates Dynamical Heterogeneity Below the Mode-Coupling Temperature

As liquids approach the glass transition temperature, dynamical heterogeneity emerges as a crucial universal feature of their behavior. Dynamic facilitation, where local motion triggers further motion nearby, plays a major role in this phenomenon. Here we show that long-range, elastically-mediated facilitation appears below the mode-coupling temperature, adding to the short-range component present at all temperatures. Our results suggest deep connections between the supercooled liquid and glass states, and pave the way for a deeper understanding of dynamical heterogeneity in glassy systems.

Dynamic facilitation is the process whereby relaxation of a local region in a glassy system enables another local region to subsequently move and relax. Facilitation plays a major role in the spatiotemporal pattern of correlated relaxation events characteristic of supercooled liquids, referred to as "dynamical heterogeneity," which is perhaps the most striking hallmark of glassy dynamics [START_REF] Berthier | Dynamical heterogeneities in glasses, colloids, and granular media[END_REF]. The underlying structural and microscopic origins of dynamic facilitation (also called kinetic facilitation) and dynamical heterogeneity are still widely debated. At high temperatures facilitation appears to be a local, shortranged process [START_REF] Fredrickson | Kinetic ising model of the glass transition[END_REF][START_REF] Chandler | Dynamics on the way to forming glass: Bubbles in space-time[END_REF][START_REF] Candelier | Spatiotemporal hierarchy of relaxation events, dynamical heterogeneities, and structural reorganization in a supercooled liquid[END_REF][START_REF] Keys | Excitations are localized and relaxation is hierarchical in glass-forming liquids[END_REF][START_REF] Elmatad | Manifestations of dynamical facilitation in glassy materials[END_REF]. However it has been appreciated for decades that some relaxation processes in glassy media involve long-ranged correlations. Given that dynamic facilitation in supercooled liquids arises initially at short time scales, where the system should behave effectively as a solid, it is natural to wonder whether long-ranged elastic processes may influence facilitation and dynamical heterogeneity.

An extreme example of the influence of elasticity is the low-temperature relaxation of tunneling defects in glasses, which is mediated by long-ranged (dipolar) phonon exchange [START_REF] Anderson | Anomalous low-temperature thermal properties of glasses and spin glasses[END_REF][START_REF] Phillips | Tunneling states in amorphous solids[END_REF][START_REF] Joffrin | Virtual phonon exchange in glasses[END_REF]. Another example is provided by the plastic behavior of amorphous solids. Local plastic deformations in elastic media induce signature quadrupolar perturbations to the stress and strain fields in the surrounding material which decay as power laws in space [START_REF] Picard | Elastic consequences of a single plastic event: A step towards the microscopic modeling of the flow of yield stress fluids[END_REF]. The role of such fields in triggering relaxation elsewhere when an amorphous solid is mechanically deformed has been the focus of intense research activity during the last decade [START_REF] Martens | Connecting diffusion and dynamical heterogeneities in actively deformed amorphous systems[END_REF][START_REF] Dasgupta | Universality of the plastic instability in strained amorphous solids[END_REF][START_REF] Gendelman | Shear transformation zones: State determined or protocol dependent?[END_REF][START_REF] Patinet | Connecting local yield stresses with plastic activity in amorphous solids[END_REF][START_REF] Budrikis | Universal features of amorphous plasticity[END_REF][START_REF] Nicolas | Deformation and flow of amorphous solids: Insights from elastoplastic models[END_REF][START_REF] Barbot | Rejuvenation and shear banding in model amorphous solids[END_REF]. This triggering by stress is a type of "elastically-mediated" dynamic facilitation that has been thoroughly studied in the context of the rheology of amorphous solids by means of elastoplas-tic models [START_REF] Nicolas | Deformation and flow of amorphous solids: Insights from elastoplastic models[END_REF], in which each rearrangement perturbs the stress of the surrounding material, thereby triggering new rearrangements when the perturbing stress surpasses some local threshold value. It should be noted, however, that elastoplastic models do not incorporate the type of thermal motion that occurs in equilibrium liquids. Recent reports of anisotropic spatial correlations in stress or strain in liquids [START_REF] Levashov | Viscosity, shear waves, and atomic-level stress-stress correlations[END_REF][START_REF] Jensen | Local shear transformations in deformed and quiescent hard-sphere colloidal glasses[END_REF][START_REF] Lemaître | Structural relaxation is a scale-free process[END_REF][START_REF] Lemaître | Tensorial analysis of eshelby stresses in 3d supercooled liquids[END_REF][START_REF] Wu | Anisotropic stress correlations in two-dimensional liquids[END_REF][START_REF] Illing | Strain pattern in supercooled liquids[END_REF] have suggested that elasticity might play a role even at high temperatures, but it is now understood that such stress correlations must arise for isotropic systems in mechanical equilibrium and do not require elasticity [START_REF] Lemaître | Inherent stress correlations in a quiescent two-dimensional liquid: Static analysis including finitesize effects[END_REF][START_REF] Lemaître | Stress correlations in glasses[END_REF].

Here we study by numerical simulation the effects of elasticity on dynamic facilitation in supercooled liquids at temperatures above and below the mode-coupling temperature, T MCT . We examine both the strain response to a rearranging particle as well as the pair correlation function of rearrangements, g hop (r), characterizing the probability of finding a rearrangement at a relative position r within a short time interval following a rearrangement at the origin. At all temperatures studied we find that the strain response of the neighborhood of a rearranging particle is anisotropic and qualitatively consistent with the response of a linearly elastic medium. The pair correlation function of rearrangements, however, reveals a change of behavior with temperature. At high temperatures, dynamic facilitation is local, i.e. it acts only on nearby regions, as indeed assumed in simple models [START_REF] Chandler | Dynamics on the way to forming glass: Bubbles in space-time[END_REF][START_REF] Garrahan | Kinetically constrained models[END_REF]. However, dynamic facilitation becomes progressively longer-ranged and elastically-mediated with decreasing temperature. Remarkably, the emergence of long-ranged facilitation takes place near T MCT , strengthening the interpretation of the mode-coupling crossover as the temperature at which the system starts to display arXiv:2103.01852v2 [cond-mat.soft] 5 Mar 2021 solid-like behaviors [START_REF] Barbot | Rejuvenation and shear banding in model amorphous solids[END_REF][START_REF] Biroli | The random first-order transition theory of glasses: A critical assessment[END_REF][START_REF] Ozawa | Role of fluctuations in the yielding transition of two-dimensional glasses[END_REF].

We conduct NVE molecular dynamics simulations, using LAMMPS [START_REF] Plimpton | Fast parallel algorithms for short-range molecular dynamics[END_REF], of a d = 2 dimensional polydisperse system of particles first introduced in [START_REF] Berthier | Zero-temperature glass transition in two dimensions[END_REF]. The swap Monte Carlo algorithm [START_REF] Berthier | Equilibrium sampling of hard spheres up to the jamming density and beyond[END_REF][START_REF] Ninarello | Models and algorithms for the next generation of glass transition studies[END_REF], allows for the equilibration of this system even well below the mode-coupling theory temperature T MCT [START_REF] Berthier | Zero-temperature glass transition in two dimensions[END_REF]. The particles have random diameters σ ∈ [σ min , σ max ] with probability density function ∝ σ -3 , where σ min and σ max are determined by the mean σ and coefficient of variation c σ of σ. Particles interact via a pair potential V (r) = GΘ(r 0r) r-12 + c 0 + c 2 r2 + c 4 r4 . Here, Θ is the Heaviside step function and c 0 , c 2 , and c 4 are chosen such that V (r 0 ) = 0 and V (r) is continuously differentiable. The argument r = r/σ is the interparticle separation r between particles i and j with diameters σ i and

σ j normalized by σ = 1 2 (σ i + σ j ) (1 -|σ i -σ j |).
Following [START_REF] Berthier | Zero-temperature glass transition in two dimensions[END_REF], we choose r0 = 2.5, = 0.2, and c σ = 0.23. We set G, σ, the (uniform) particle mass m, and the Boltzmann constant k B to unity, thus defining our energy, length, mass, and temperature units. We study N = 10 4 particles in an L × L periodic square box of side length L = 100. For this system the onset temperature is T 0 ≈ 0.236 and the mode-coupling temperature is T MCT ≈ 0.116 [START_REF] Berthier | Zero-temperature glass transition in two dimensions[END_REF].

We study rearrangements in 400 trajectories with independent initial velocities chosen from the Boltzmann distribution for each of 202 separate equilibrium configurations, for eight temperatures T = 0.100, 0.105, 0.110, 0.115, 0.120, 0.130, 0.140, and 0.150, doubling our statistics by averaging over trajectories in both directions in time [33]. We identify rearranging particles by focusing on quenched trajectories between inherent structures, and by measuring the squared displacement ∆r 2 between inherent structures [START_REF] Lemaître | Structural relaxation is a scale-free process[END_REF]33]. Our criterion for rearrangement is ∆r 2 > 5a 2 across a time interval ∆t = 10 2 , where a is the plateau height of the root mean squared displacement for particles of similar size in the unquenched system at the given temperature [33]. The time scale ∆t = 10 2 matches the time at which dynamic facilitation is at play, identified as the one at which individual rearrangements cluster together in space [33]. For reference, the ballistic time is τ ballistic ≈ 10 0 in these units, and the relaxation time is τ α = 3 × 10 2 and τ α = 2 × 10 6 at the highest and lowest temperatures, respectively. For further characterization and details on the choice of ∆t, see the Supplementary Material [33].

Our first observation is that the rearrangement of a given particle causes elastic-like displacements of other particles; i.e. the system, despite being a liquid, displays a solid-like response at short times. To show this, we proceed as follows. For each rearrangement in the system, we obtain the local best-fit linear strain tensor E = E(r) for all particles from the symmetrized local quenched displacement gradient tensor [START_REF] Jensen | Local shear transformations in deformed and quiescent hard-sphere colloidal glasses[END_REF]33,[START_REF] Falk | Dynamics of viscoplastic deformation in amorphous solids[END_REF]. We then extract strain invariants, such as the isotropic strain b,d,f). Here, E is the best-fit linear strain tensor at r, E is detraced E, and eext is a unit eigenvector corresponding to the largest eigenvalue of E at the origin. The radial axes are plotted on a log scale.

γ iso = TrE and the deviatoric strain

γ dev = √ 2TrE 2 (where E ≡ E -1 d I TrE).
Crucially, E also gives us local extensional and compressional axes from its orthonormal eigenvectors: we denote as e ext (e com ) the eigenvector corresponding to the largest (smallest) eigenvalue of E. We then average over neighborhoods of rearranging particles translated such that the rearranging particle is at the origin at the start of the rearrangement interval (of duration ∆t) and rotated such that the extensional axis of its local strain tensor is e ext = (1, 0). We also calculate the strain in this direction, γ ext ≡ E : e ext ⊗ e ext , where e ext and e com are eigenvectors of the local strain tensor E computed at the rearranging particle i, whereas E is the local deviatoric strain tensor of the neighbors j of i. The dipolarity of γ iso , quadrupolarity of γ ext , and the isotropic behavior of γ dev , shown in Fig. 1 for temperatures T = 0.100 (left) and T = 0.150 (right) spanning the full temperature range of our study, are the expected signatures of the response of an elastic solid [START_REF] Zhang | Interplay of rearrangements, strain, and local structure during avalanche propagation[END_REF].

We now assess the temperature dependence of the range of the elastic-like response by studying the angular Fourier series coefficients, which characterize the strength of the observed symmetries reported above [33]. More specifically, we consider the zeroth and second order coefficients γiso,0 and γiso,2 , the zeroth order coefficient γdev,0 and the fourth order coefficient γext,4 . As shown in Fig. 2, γiso,2 and γext,4 (Fig. 2(c,d)) decay for T ≤ 0.12 as r -2 from r = 3 up to a cutoff that increases slightly with decreasing T but is roughly r 10. This power-law decay is expected for an elastic response [START_REF] Zhang | Interplay of rearrangements, strain, and local structure during avalanche propagation[END_REF]. The radial dependence of γdev,0 (Fig. 2(b)) deviates from the elastic response behavior at higher temperatures but appears to approach the expected behavior (dashed line) with decreasing temperature. This can be explained by the decreasing density of rearrangements per unit time with decreasing T , which allows the elastic displacement field from rearrangements to extend further into the material without being obscured by other rearrangements [33]. Finally, for the zeroth mode of γ iso (Fig. 2(a)), steadystate elasticity predicts γiso,0 ≡ 0. Instead, we see a rapidly-decaying radial wave of isotropic expansion and contraction, with expansion peaks (compression troughs) roughly coinciding with the peaks (troughs) of g (r) [33]. This is consistent with the results of [START_REF] Zhang | Interplay of rearrangements, strain, and local structure during avalanche propagation[END_REF] in the context of athermal quasistatic shear: rearrangements expand the structure at the shortest distances from the rearrangement, thus locally hardening the structure, but compress the structure slightly further out, with this hardeningsoftening pattern repeating out to large distances.

We now focus directly on dynamic facilitation and analyze its relation with the elastic responses characterized above. We use the pair distribution of rearrangements, g hop (r) = g hop (r, θ) (the average density of rearrangements at r given a rearrangement at the origin, divided by the bulk rearrangement density [33]), to investigate the extent to which facilitation of rearrangements is due to elasticity. In Fig. 3(a,b) we display ghop (r, θ) = 2πg hop (r, θ)/ 2π 0 g hop (r, θ)dθ (the rearrangement pair distribution g hop (r, θ) normalized at each r by its angular mean 1 2π 2π 0 g hop (r, θ) dθ) for rearrangements over the time interval [0, ∆t] [36] at distances r = 1, 3, 10, 30, for temperatures T = 0.100 and T = 0.150. Well below T MCT , at T = 0.100 (Fig. 3(a)), the spatial distribution of subsequent rearrangements remains anisotropic out to r = 30. The angular Fourier series expansion coefficients of ghop [33] show that this anisotropy comes from the second-and fourth-order Fourier modes of ghop , and persists up to at least r ≈ 20. Above T MCT , at T = 0.150 (Fig. 3(b)), no anisostropic structure can be discerned beyond the second neighbor shell, i.e. beyond r > 3.

To quantify the amount of anisotropy in the spatial distribution of rearrangements we introduce the mean squared anisotropy (deviation from isotropy) [START_REF]When calculating α in practice, we control for effect of statistical noise, which is larger at lower temperatures due to these temperatures having[END_REF]

α(r) = 1 2π 2π 0 (g hop (r, θ) -1)
2 dθ, as shown in Fig. 3(c). High α at small r shows that displacements at this scale, though highly nonlinear, are measurably influenced by the orientation of the linear strain. This short-ranged influenceindicative of short-ranged dynamic facilitation-persists over the entire temperature range studied. As T is lowered, the large-r tail of α increases markedly, spanning two orders of magnitude, implying a significant difference in the role played by long-range elastic interactions between the lowest and highest temperatures considered here. At the same time, Fig. 3(d) shows that the pair correlation for hops, g hop (r), becomes progressively longerranged with decreasing temperature, indicating that rearrangements are triggered further and further away.

To quantify the extent of elastically-mediated dynamic facilitation, we introduce I α := 40 2.5 α(r)dr, which integrates α over distances beyond the second neighbor shell at r 2.5 (to filter out the effect of short-range facilitation) and up to the limit beyond which finitesize effects appear. Fig. 4(a) displays its evolution with 1/T . We observe a crossover at T MCT from a slow to a rapid growth of I α as a function of 1/T . Above T MCT , the strain-correlated spatial organization of the rearrangements is largely obscured, beyond the secondneighbor shell, by other rearrangements [33]. The competition between these other rearrangements and emergent elasticity at longer distances leads to the observed crossover in I α . Note that the observed crossover at T MCT is robust to choices of lower bound of the integral, rearrangement interval ∆t, strain length ξ FL , and starting point for the integration (r 0 ≥ 2.5) [33]. We similarly define an integrated excess density of rearrangements (relative to the bulk rearrangement density), I hop := Taken together, our findings suggest a crossover from solely short-ranged dynamic facilitation for T > T MCT to additional long-ranged, elastically-mediated dynamic facilitation of avalanches at T < T MCT . This finding provides an intriguing link between mean-field models, where T MCT marks the location where supercooled liquids first exhibit a transient solid-like behavior, and real-space dynamical heterogeneity. We note that the crossover at T MCT is not visible in the strain field induced by a rearrangement; for example, the isotropic strain shows dipolar angular behavior at all temperatures considered. The crossover is only evident in the spatial and angular distribution of rearrangements that follow soon after the original rearrangement at the origin. The anisotropic spatial distribution of hops evidenced in g hop , Fig. 3(a,b), and the increase in range of g hop (r), Fig. 3(d), as well as the crossover at T = T MCT seen in I α , Fig. 4(a), can only be due to the strain field created by the rearrangement at the origin. At high T strain does not facilitate rearrangements because the rearrangements are so abundant and closely spaced that their strain fields effectively interfere [33]. We note that at temperatures below T MCT , dynamical heterogeneity still involves short-ranged facilitation, whereby the local relaxation of clusters coalesces on longer length scales to form avalanches. Nevertheless, the emergence of prominent long-ranged elastic effects must influence the spatiotemporal correlations between such avalanches.

Approximately ten years ago it was noticed that rare, long-ranged "surges" of dynamical heterogeneity connected to strain deformation occur in supercooled liquids [START_REF] Widmer-Cooper | Central role of thermal collective strain in the relaxation of structure in a supercooled liquid[END_REF], and that such behavior occurs in some coarsegrained kinetically-constrained models [START_REF] Chandler | Dynamics on the way to forming glass: Bubbles in space-time[END_REF][START_REF] Keys | Excitations are localized and relaxation is hierarchical in glass-forming liquids[END_REF]. Our findings potentially pinpoint the microscopic underpinnings of such behavior, which could lead to greater understanding of the building blocks of phenomenological models of supercooled liquids. It would be certainly worthwhile to generalize elastoplastic models [START_REF] Nicolas | Deformation and flow of amorphous solids: Insights from elastoplastic models[END_REF] to describe elastic dynamic facilitation within equilibrium thermal dynamics along the lines of [START_REF] Popović | Thermally activated flow in models of amorphous solids[END_REF][START_REF] Parley | Aging in a mean field elastoplastic model of amorphous solids[END_REF]. More generally, our results highlight the importance of long-ranged elastic processes in mediating the organization of dynamical heterogeneities on long length and time scales, and the need to incorporate such processes in simplified models of the glass transition.

We thank M. Ozawa and L. Berthier for providing us with equilibrated configurations and the code to generate them, C. Scalliet for providing the swap potential

FIG. 1 .

 1 FIG.1. Polar plots of (a,b) γiso = TrE, (c,d) γ dev = √ 2TrE 2 , and (e,f) γext = E : eext ⊗ eext across a rearrangement interval ∆t = 10 2 as a function of the position r relative to a rearranging particle that was at r = 0 at the start of the rearrangement interval for systems at temperatures T = 0.100 (a,c,e) and T = 0.150 (b,d,f). Here, E is the best-fit linear strain tensor at r, E is detraced E, and eext is a unit eigenvector corresponding to the largest eigenvalue of E at the origin. The radial axes are plotted on a log scale.

4 FIG. 2 .

 42 FIG. 2. (a-d):Fourier series cofficients γiso,0, γdev,0 , γiso,2, and γext,4, respectively, as a function of distance r from a rearranging particle (c.f. Fig.1). Dotted segments denote negative values. The colors correspond to temperatures T = 0.100, 0.105, 0.110, 0.115, 0.120, 0.130, 0.140, and 0.150 (darkest lightest). Dashed lines (b-d): Slopes corresponding to the power-law decay, ∝ r -2 , of an infinite elastic medium responding to a small plastic deformation. Dashed line (a): Power law decay ∝ r -3 plotted as an upper bound to the envelope of γiso,0.

FIG. 3 .

 3 FIG. 3. Top row: Plots of normalized pair distribution of rearrangements ghop = 2πg hop / 2π 0 g hop dθ across one rearrangement interval ∆t = 10 2 for temperatures (a) T = 0.100 and (b) 0.150. Each curve corresponds to a different r, and is scaled so that isotropy (dashed black circle) corresponds to the corresponding r value for the log-scaled r of Fig. 1. (c): Mean squared anisotropy α(r) = 1 2π

40 2. 5

 405 2πr (g hop (r) -1) dr, where g hop (r) = 1 2π 2π 0 g hop (r, θ)dθ. Fig.4(b)shows that I hop increases with decreasing T , showing that hops trigger increasingly distant hops. These results for I α and I hop provide direct and explicit evidence that the elastic response to a rearrangement triggers other, distant rearrangements, demonstrating the emergence of elastic dynamic facilitation when crossing T MCT .

40 2. 5

 405 FIG. 4. Integrated (a) mean squared anisotropy Iα = 40 2.5 α(r)dr and (b) pair distribution of rearrangements I hop = 40 2.5 2πr(g hop (r) -1)dr as a function of inverse temperature 1/T . Dotted lines: 1/TMCT.

files for use with LAMMPS, and E. Corwin for sharing computational resources with us in the early stages of this project. RNC thanks S. Ridout, G. Zhang, and I. Tah for discussions. This work received funding from the Simons Collaboration "Cracking the glass problem" via 454935 (G. Biroli), 454945 (A. J. Liu), 454951 (D. R. Reichman) and 348126 (R. N. Chacko), and from a Simons Investigator grant (327939 to A. J. Liu). This work used the Extreme Science and Engineering Discovery Environment (XSEDE) [41], which is supported by National Science Foundation grant number ACI-1548562.