Abdelhak Loukkal

Vincent Fremont

Yves Grandvalet

You Li

Improving semantic segmentation in urban scenes with a cartographic information

This paper presents three different approaches to inject a location information in semantic segmentation Convolutional Neural Networks (CNN) applied to urban scenes. The assumption that a location information would improve semantic segmentation performance emerges from the idea that some elements of urban scenes are located in a predictable manner. This assumption is confronted to realistic data on the CARLA autonomous driving simulator, which is used to create our own synthetic dataset with images, depth maps and bird-eye-view cartographic images. Simulators circumvent the difficulties due to the scarcity of publicly available synchronous labeled images and location information. We consider the location information as a cartographic image as we assume it is the simplest option to include it in a CNN. We assess the relevance of injecting the cartographic information in three different manners: as a CRF potential, as an additional task and as an additional encoder input of a CNN. The three methods are evaluated and compared with a state of the art CNN with regards to the pixel-wise accuracy, mean intersection over union and intersection over union of some important classes. The multi-encoder approach improves the intersection over union of the pedestrians, vehicles and traffic signs classes by respectively 4%, 1.6% and 9 %.

I. INTRODUCTION

Semantic segmentation has been drawing a lot of attention from the computer vision and autonomous driving communities for many years because in addition to detecting key elements in the scene, it adds semantic information to the global scene understanding problem.

Urban scenes are very challenging environments for autonomous driving because of the dynamics of observed objects. However, some objects' locations in these scenes are quite predictable: for example it is impossible to have a building in the middle of a road and cars are more likely to be found on the road rather than on the sidewalk. The intuition about using a location information in the form of a digital high definition map in perception modules seems reasonable. The objective of this paper is to prove that adding the cartographic information to a semantic segmentation neural network can improve its accuracy regarding some important class of objects like vehicles, pedestrians or traffic signs.

For a neural network to be able to take advantage of some cartographic information, it would be necessary to have a dataset containing labeled images with their corresponding cartographic information. In the publicly available datasets for semantic segmentation, GPS data is available but we lack a precise cartographic information. For this reason, we choose to work on synthetic data produced by CARLA autonomous driving simulator [START_REF] Dosovitskiy | CARLA: an open urban driving simulator[END_REF] which provides a complete platform with semantic segmentation, depth, LIDAR and a precise map of a virtual town.

For each frame obtained from the simulator, we extract the part of the map that corresponds to one hundred meters ahead of the vehicle. These extracted portions of the map come in a bird-eye-view configuration. We apply an inverse perspective mapping to project the map portion in the camera plane, so as to ease matching with camera frames. We propose three neural networks designs for processing this cartographic information in a CNN. First, we add the map as a pairwise potential in a CRF on top of a state of the art CNN. Second, we design a multi task network that outputs semantic segmentation, depth and the cartographic map. Finally, we add it as an additional input in a network with multiple encoder streams.

II. RELATED WORK

The success of deep CNNs for image classification [START_REF] Krizhevsky | Imagenet classification with deep convolutional neural networks[END_REF] has encouraged researchers to explore the effectiveness of these networks for dense predictions tasks like semantic segmentation or depth estimation. The current state of the art approaches in semantic segmentation leverage the idea of fully convolutional neural networks [START_REF] Shelhamer | Fully convolutional networks for semantic segmentation[END_REF] keeping the spatial information, that is usually lost in regular CNNs, by avoiding fully connected layers. These networks come in two parts: the encoder extracts features from the input image and the decoder up-samples the encoded feature map to match the size of the input image. State of the art networks for pixel wise semantic segmentation take also advantage of spatial pyramid pooling [START_REF] He | Spatial pyramid pooling in deep convolutional networks for visual recognition[END_REF] and dilated convolutions [START_REF] Yu | Multi-scale context aggregation by dilated convolutions[END_REF] to segment objects at different scales and enlarge the receptive fields, hence the context, of the convolution filters.

Semantic segmentation is a very complex task for which satisfying results were obtained thanks to public datasets like KITTI [START_REF] Geiger | Vision meets robotics: The KITTI dataset[END_REF], CamVid [START_REF] Gabriel | Semantic object classes in video: A high-definition ground truth database[END_REF] and Cityscapes [START_REF] Cordts | The cityscapes dataset for semantic urban scene understanding[END_REF] which contains much more labeled images than the two previous ones. More recently, thanks to a new labeling pipeline, Apolloscapes [START_REF] Huang | The apolloscape dataset for autonomous driving[END_REF], a huge dataset of more than 100k labeled images was released.

gm random fields (CRF) are graphical models that are commonly used for semantic segmentation. Their energy function is composed of unary potentials and pair-wise potentials, see [START_REF] Krizhevsky | Imagenet classification with deep convolutional neural networks[END_REF], on neighbors (can be pixels or patches of pixels). Originally, the main limitation of this model was its inability to capture long range dependencies, with pixels that are in different regions of the image. Region based approaches that incorporate hierarchical connectivity and higher-order potentials tried to improve on the original CRFs but lacked accuracy due to the unsupervised segmentation that produces regions. Fully connected CRFs are more expressive models, which present the advantage to have pair-wise potentials between all pixels in an image but the difficulty of inference hindered their potential. A mean field algorithm was proposed to learn efficiently fully connected CRFs [START_REF] Krähenbühl | Efficient inference in fully connected crfs with gaussian edge potentials[END_REF]. CRFs have been successfully applied as a post processing or refinement step in CNNs in order to improve boundaries delineation for segmentation tasks. [START_REF] Zheng | Conditional random fields as recurrent neural networks[END_REF] proposed an approach that includes the CRF in an end-to-end CNN-CRF where the Mean Field approximate inference is formulated as a recurrent neural network and the Mean Field algorithm steps reformulated as convolution operations.

Autonomous driving involves different tasks like semantic segmentation, object detection or depth estimation. Having a single neural network that solves all these tasks is an important asset. Multi-task learning is a learning strategy that consists in learning different tasks simultaneously, sharing knowledge between tasks. For example, a shared representation improves learning efficiency and prediction accuracy. Multi-task learning is an interesting framework for autonomous driving and mobile robotics in general because it combines different systems in one and eases real-time computing. The multi-task learning loss is a weighted sum of the tasks' individual losses. This weighting can be either uniform or tuned manually. Recently, [START_REF] Kendall | Multi-task learning using uncertainty to weigh losses for scene geometry and semantics[END_REF] provided a new weighting approach based on homoscedastic uncertainty, an uncertainty that captures the relative confidence of the different tasks.

Dense prediction networks used for semantic segmentation and depth estimation come in the encoder-decoder configuration. Using several streams for the encoder has been studied in several papers, among the first is [START_REF] Hazirbas | Fusenet: Incorporating depth into semantic segmentation via fusionbased CNN architecture[END_REF] that fuses the features extracted from the image and the depth at different scales in the encoder to improve semantic segmentation.

Using a contextual information to improve semantic segmentation has been explored in previous papers. [START_REF] Kohli | Robust higher order potentials for enforcing label consistency[END_REF] augment pair-wise CRFs potentials with higher order potentials defined on sets of pixels that are determined with unsupervised segmentation algorithms. [?] More recently [START_REF] Wang | Semantic segmentation of urban scenes with a location prior map using lidar measurements[END_REF] introduced an approach where street layouts are estimated then used to compute spatial prior maps that are used as additional potentials in a CRF to improve semantic segmentation.

III. SYNTHETIC DATASET

Real world semantic segmentation datasets are very expensive to annotate and large-scale public datasets do not provide precise cartographic information. Therefore, simulation was the designated solution for our problem. Among the available autonomous driving simulators [START_REF] Shah | Airsim: High-fidelity visual and physical simulation for autonomous vehicles[END_REF], [START_REF] Best | Autonovi: Autonomous vehicle planning with dynamic maneuvers and traffic constraints[END_REF], [START_REF] Dosovitskiy | CARLA: an open urban driving simulator[END_REF], we chose CARLA as image, depth, semantic and cartographic map are easy to access. The map of the virtual town is provided as a PNG image encoded in three layers with one layer giving information about roads, another one about intersections and the last one about lanes. The car can be positioned in the 2D image map through its world location, using the transformation (provided by the authors) from world coordinates to pixel coordinates in the 2D image map.

All the methods investigated in this paper rely on the fact that the cartographic map is projected in the camera plane. So first, for each camera frame extracted from the simulator, using the coordinates of the vehicle, a portion of the image map corresponding to one hundred meters ahead of the vehicle is extracted. Then, using the transformation between the bird-eye-view plane and the camera plane, the points of the bird's-eye view map are projected in the camera plane with inverse perspective mapping, see Fig. 1. The transformation between the two planes is the following:

p ∼ K[R|T ]w (1)
where K is the intrinsic matrix of the camera, R the rotation matrix, T the translation matrix, p = (u, v, 1) T the coordinates in the camera plane and w = (x, y, 1) T the coordinates in the bird-eye-view plane. This projected map is homogeneous to a segmentation label. We assign labels to road, intersection and "other" pixels. Alignment of the projected map and the road in images is not perfect because of the vehicle dynamics that change the rotation and translation matrices that define the inverse perspective mapping. This could be compensated using the vehicle's inertial sensors but we chose not to do it to be more realistic.

IV. PROPOSED METHODS

Our approach is based on the idea that the cartographic image is considered as a prior to inject in a CNN. Here we use Deeplab V2 [START_REF] Chen | Deeplab: Semantic image segmentation with deep convolutional nets, atrous convolution, and fully connected crfs[END_REF], a state of the art network for pixelwise semantic segmentation. We have explored three ways of incorporating the cartographic information in the network: Fig. 2: Method 1: Different combinations of pairwise potentials have been tried to improve the semantic segmentation results. Trials using the cartographic map and the depth map as separate appearance kernels and a single kernel based on the element-wise fusion of depth and cartographic map have been performed.

• The first method consists in adding the ground truth cartographic and depth maps as additional entries in the pairwise potentials of the CRF-RNN layer on top of a CNN (see Fig. 2). • The second method is similar to the previous one regarding the CRF except that the depth and cartographic maps that are used for the CRF are predicted by a multi task network instead of being ground truth (see Fig. 3). • The last method is a CNN with three encoder streams: image, depth and cartographic maps. The features extracted from the three streams are fused by element-wise operations and fed to a decoder that outputs a semantic segmentation map (see Fig. 4).

A. Deeplab with CRF-RNN layer

We design our network using Deeplab V2 as our basis. This network takes advantage of dilated convolution to enlarge the size of the feature maps and the receptive field without increasing the number of parameters and a trous spatial pyramid pooling to aggregate multi scale information. We add an additional CRF-RNN layer at the output of the network to allow end-to-end training of the CRF with the CNN. The Gibbs energy of the fully connected CRF is the following:

E(x) = N i=1 ψ u (x i ) + i<j ψ p (x i , x j ) (2)
The unary potential ψ u (x i ) is computed for each pixel by the CNN that produces a distribution over the label assignment x i . The pair wise potentials are a linear combination of Gaussian kernels:

ψ p (x i , x j ) = µ p (x i , x j ) K m=1 w (m) k (m) (f i , f j ) k(f i ,f j ) (3)
where k (m) is a Gaussian kernel, w (m) a weight, f i a feature vector and µ a compatibility function that can be given by a simple potts model

µ p (x i , x j ) = [x i = x j ].
The original paper on fully connected CRFs [START_REF] Krähenbühl | Efficient inference in fully connected crfs with gaussian edge potentials[END_REF] defines the following pairwise potentials that have been successfully applied since:

k(f i , f j ) = w (1) exp(- |p i -p j | 2 2θ 2 α - |c i -c j | 2 2θ 2 β ) appearance kernel + w (2) exp(- |p i -p j | 2 2θ 2 γ ) smoothness kernel (4)
where c i and p i are respectively the vector of RGB values for pixel i and the position in the image.

We explore here different combinations of potentials. We consider adding the cartographic map and the depth map as separate appearance kernels and we also evaluated adding a kernel based on the fusion of both depth and cartographic map that we call focus map:

k f ocus (f i , f j ) = exp(- |p i -p j | 2 2θ 2 α - |F i -F j | 2 2θ 2 β ) (5)
where:

F = (1/D) * M ( 6
)
and F is the obtained 2D focus map, D the 2D depth map, M the 2D cartographic map (analogous to a segmentation map with label 2 for intersections, label 1 for the road and label 0 everywhere else) and the division and multiplication are element-wise operations.

The weights of the different Gaussian kernels are learned end to end through back-propagation of the gradients in the network.

B. Multi task network

To design the multi task network, we built it on Deeplab to which we add two decoder branches, one for the depth estimation and one for the map estimation. The Deeplab Fig. 3: Method 2: Multi task network with semantic segmentation, cartographic map and depth outputs. The output of the semantic segmentation branch is used as a unary potential in the CRF layer and depth and map outputs are used as pair-wise potentials in the CRF. network outputs feature maps eight times smaller than the original input image, so bilinear sampling is used for upsampling. The map estimation being considered as another segmentation task, the decoder we use is similar to the one used in the original Deeplab. For the depth estimation task, we added 3 up-convolution layers to resize the output to the input size and obtain a smoother depth estimation. To weight the different losses, we did manual tunning of the weights. The best results were obtained with the following configuration:

Loss total = 10 * Loss S + 0.1 * Loss D + Loss M (7)
where Loss S , Loss D and Loss M are respectively the loss functions for semantic segmentation, depth estimation and map segmentation. This particular weighting was obtained empirically.

The loss function for the semantic segmentation and cartographic map estimation tasks is the cross entropy loss. For the depth estimation task, we have chosen the BerHu loss which was shown to yield better results than the L1 loss [START_REF] Laina | Deeper depth prediction with fully convolutional residual networks[END_REF]. The reverse Huber loss is defined as:

B(x) = |x| if |x| ≤ c x 2 +c 2 2c if |x| ≥ c (8)
For every gradient descent step where we compute B(y -ŷ), c is defined as:

c = 1 5 max i (| ŷi -y i |) (9)
where y i and ŷi are respectively the ground truth depth and predicted depth. c is in other words twenty percent of the maximal per batch error. This loss function is equivalent to L1 loss when x ∈ [-c, c] and to L2 loss otherwise. We add the CRF-RNN layer at the end of the semantic segmentation branch. The output of the depth estimation and cartographic map estimation branches are then used as additional pair-wise potentials in the CRF, see [START_REF] Shelhamer | Fully convolutional networks for semantic segmentation[END_REF].

C. Multi encoder streams network

In this section, we use a neural network with multiple encoder streams. We build from Deeplab network to which we add two encoder streams, one for the depth and another one for the map. The feature maps extracted from the three streams are fused to obtain one feature map that is passed to the semantic segmentation decoder. The fusion strategy that achieved the best results is done first by element-wise multiplication of the image and map features, then the result of the multiplication is added element-wise to the depth features.

V. EXPERIMENTS

We run the experiments on a set of two NVIDIA Titan X. We use ADAM optimizer [START_REF] Diederik | Adam: A method for stochastic optimization[END_REF] with an initial learning rate of 10 -4 and exponential decay. We compare the three networks with the original Deeplab without CRF. We compare with different CRF pair-wise potentials and encoder streams to highlight the impact of the cartographic map. All networks are fine-tuned from the pretrained solution fitted on Cityscapes and Pascal VOC. All the networks containing the CRF-RNN layer were trained with a batch size of one because of implementation limits of the custom CRF-RNN layer, other networks were trained with a batch size of three , due to limited computational resources. In this paper, Tensorflow has been used to design and train the CNNs. For the CRF part of the code, the keras/tensorflow code given by [START_REF] Zheng | Conditional random fields as recurrent neural networks[END_REF] was used. The metric used to evaluate the performance of the network is the IoU, intersection over union, defined as:

IoU = T P T P + F P + F N
where TP, FP and FN are respectively the true positive, false positive, and false negative pixel counts on the set of test images. This metric is evaluated for each class and the mean over all classes is computed as a general summary.

A. Dataset

We generated 4700 labeled images using the CARLA autonomous driving simulator. We ran 10 episodes of 470 frames each, each episode starting from a different position in the virtual world. We used virtual town number one for all our experiments. The number of vehicles in the world

B. Deeplab with CRF-RNN layer

We have first fine-tuned a pretrained Deeplab V2 network on our dataset for 10 epochs. We train a Deeplab network with an additional CRF-RNN layer and compare the results with those of the regular Deeplab network that was finetuned on the training data. We tested different additional pairwise potentials. The results are shown in Table I. We observe that the CRF with the cartographic map and depth map as separate additional potentials has the best results regarding the fences and traffic signs IoU improving the IoU by respectively 2.2 % and 3.3% compared to the regular Deeplab and the focus map additional potential has the best results for the vehicles, pedestrians and poles IoU improving the IoU by respectively 0.9% , 3.9% and 0.5 %. This gives a hint on the fact that adding the cartographic information in the CRF adds an information on the location of some important classes like pedestrians, vehicles or traffic signs that are more likely to be found respectively on or outside the road. The Deeplab network with CRF and no additional potential has the best overall performance regarding the classes of interest and achieves the best mIoU and pixel wise accuracy and the best road and road lines IoUs.

C. Multi task model

For the multi task model, we have used the pretrained weights for the encoder and trained the network for ten epochs first. Once the network has learned to predict depth and cartographic map, the outputs of these two branches have been injected in a CRF-RNN layer at the end of the semantic segmentation branch and the network was trained for ten more epochs. Therefore, results with the different CRF pairwise additional potentials are compared with the regular Deeplab and the multi task network without CRF both trained during 20 epochs. The results are shown in Table II. The multi-task network is referred to as MT for brevity. The networks with additional pair-wise potentials are referred to as M T + CRF + additional -potentials. The CRF with the focus map, here the focus map is obtained by fusing the output of the depth and map branches of the network, has the highest mean IoU out performing the regular Deeplab by 1.2 % but regarding the classes of interest, the regular Deeplab has the highest IoUs. When comparing the multi task network with and an without CRF, we observe that in this case, the CRF doesn't improve the results in the main classes of interest.

D. Multi encoder model

For this approach, we have used the pretrained weights for the image branch and trained from scratch the depth and map branches. We have tested different fusion strategies and different fusion combinations. The best results were obtained by multiplying the image features by the cartographic map features and then adding the depth features. The results are shown in Table III. The multi-encoder network is referred to as ME to simplify notation. The networks with additional encoders are referred to as M E : Operations. This method achieves the best results among the three tested methods and even in ten epochs has better results in the important classes than the CRF, regular Deeplab and multi task approaches in respectively ten, twenty and twenty epochs. The multi stream network where we multiply by the map and add the depth improves the IoU of the pedestrians, vehicles and traffic signs classes by respectively 4%, 1.6% and 9 %. Multiplying by the features of the cartographic map insures a stronger relationship with the features of the image and enforces the relation between the classes and there location in the map.

E. Discussion

This paper confirms the idea that the predictability of some objects' location in a road scene can help improving the semantic segmentation and presents encouraging results regarding the use of a cartographic information as an image in CNNs. We hope that this line of works will encourage the release of publicly available real world datasets with synchronous semantic segmentation labels and precise cartographic information. The maps would need to have a high precision at centimeter-level. This kind of maps is already popular for autonomous driving and is called High Definition maps. In this paper, the only available information in the map is about the road boundaries and intersections. HD maps contain richer information: about the lanes which can potentially improve the accuracy of the segmentation even more, especially road lines IoU. Whatever format these HD maps come in, it would be possible to rasterize them to use them as proposed in this paper.

VI. CONCLUSION

The location information yields important information about a vehicle surroundings. In this paper, we have investigated how to inject a location information in a CNN, considering it as another image input to the network. We have explored adding the map image as a pair wise potential in a CRF, as an additional task in a multi task framework and as an additional encoder branch. After comparison with the regular Deeplab network, adding a map based pair-wise potential in the CRF can improve slightly the intersection over union of important classes but the regular CRF still has a better overall performance. Training a multi task network with a cartographic map prediction branch fails to improve the performance regarding the most important classes. Finally, adding the map information as an additional encoder branch of a segmentation network insures the best results improving significantly the intersection over union of important classes like vehicles, pedestrians or traffic signs.

Fig. 1 :

 1 Fig. 1: Inverse perspective mapping pipeline

Fig. 4 :

 4 Fig. 4: Method 3: Multi encoder stream network. The features extracted from the depth and map encoders are fused, by element wise multiplication for the map and element wise addition for the depth, with the features extracted from the RGB image.

TABLE I :

 I Results of the first method. Best results are shown in bold. All networks are trained on 10 epochs.

		Deeplab Deeplab + CRF Deeplab + CRF + depth Deeplab + CRF + depth & map Deeplab + CRF + focus map
	mIoU	50.2	52.4	48.2	50.5	51.1
	Pixel accuracy	88.3	88.8	86.5	87.8	88.6
	Vehicle IoU	86.3	87.00	86.3	86.5	87.2
	Pedestrians IoU	18.0	20.7	16.6	19.8	21.9
	Traffic sign IoU 57.0	60.2	57.2	60.3	60.0
	Fences IoU	37.1	38.7	34.7	39.3	39.3
	Poles IoU	29.7	30.2	28.0	30.1	30.2
	Road IoU	89.7	90.7	88.6	89.2	89.7
	Road lines IoU	22.6	39.4	35.2	36.5	24.9

TABLE II :

 II Results of the second method. Best results are shown in bold. Regular Deeplab and MT without CRF are trained on 20 epochs, the MT CRF networks are trained first on 10 epochs without CRF then on another 10 epochs with CRF.

		Deeplab MT	MT + CRF MT + CRF + depth MT + CRF + depth & map MT + CRF + focus map
	mIoU	52.1	52.3 53.2	52.4	51.9	53.3
	Pixel accuracy	88.8	88.8	88.5	88.2	88.4	88.7
	Vehicle IoU	87.4	87.3 87.2	87.0	87.1	86.9
	Pedestrians IoU	20.8	18.9 19.0	20.0	19.3	18.9
	Traffic sign IoU 61.7	63.3 62.2	64.2	63.4	63.5
	Fences IoU	39.4	38.4 37.7	38.5	38.2	38.5
	Poles IoU	30.6	30.1 30.4	29.7	30.0	30.2
	Road IoU	89.9	89.9	90.8	89.5	89.8	89.3
	Road lines IoU	22.4	23.4	39.0	29.6	22.5	35.4

TABLE III :

 III Results of the third method. Best results are shown in bold. The networks evaluated in this table are trained on 10 epochs

		Deeplab ME: Image + depth ME: Image + focus map ME: Image * map ME: Image* map + depth
	mIoU	50.2	51.6	50.7	49.6	50.7
	Pixel accuracy	88.3	88.5	88.6	88.55	88.6
	Vehicle IoU	86.3	87.5	87.5	87.6	87.9
	Pedestrians IoU	18.0	19.7	20.9	20.6	22.0
	Traffic sign IoU 57.0	63.5	62.8	62.6	66.0
	Fences IoU	37.1	40.4	39.8	38.3	39.9
	Poles IoU	29.7	30.2	30.1	30.1	30.6
	Road IoU	89.7	89.8	89.93	90.0	90.0
	Road lines IoU	22.6	23.1	21.7	22.1	23.5

ACKNOWLEDGEMENT

This work has been conducted within SIVALAB, a joint research laboratory between Renault and Heudiasyc, which targets issues pertaining to integrity questions in autonomous driving.


S Medrano

H Zhao

F De Geuser

B Gault

L T Stephenson

A Deschamps

D Ponge

D Raabe

C W Sinclair

Cluster hardening in Al-3Mg triggered by small Cu additions

Keywords: Precipitation, aluminum alloys, yield strength, atom probe

The aging response of two Al-3Mg alloys with Cu addition <1 wt% has been tracked under simulated automotive paint bake conditions (⇠ 20 min, 160 and 200 C) to quantify the processes controlling hardening. The decomposition of the solid solution, observed by atom probe tomography, has been interpreted using a novel pair correlation function approach and incorporated into a model for prediction of precipitation hardening. It is shown that the hardening is controlled by clusters/Guinier-Preston-Bagaryatsky (GPB) zones similarly to what has been previously observed in much higher Cu containing 2XXX-series alloys. Interestingly, it is shown that very small additions of Cu (< 0.1at%) can be used to catalyze a high number density of strengthening particles owing to the high enrichment in Mg compared to particles found in more conventional high Cu/low Mg alloys. This allows hardening during the first hour of aging that is as high as that obtainable in these high Cu alloys.

Introduction

There is evidence that additions of less than 0.5at%Cu to 5XXX-series aluminum alloys can counterbalance the softening experienced by stamped then 'paint baked' (⇠ 150-200 C for ⇠ 20 min) automotive aluminum components [START_REF] Court | The Ageing and Thermal Recovery Behaviour of Al-Mg-Cu Alloys[END_REF][START_REF] Ratchev | Artificial Ageing of Al-Mg-Cu Alloys[END_REF][START_REF] Engler | Influence of copper additions on materials properties and corrosion behaviour of AlMg alloy sheet[END_REF]. Being able to retain the as-stamped strength in this way facilitates both reduced material usage and adoption of lower grade Cu-residual scrap in primary alloy production [START_REF] Das | Emerging Trends in Aluminum Recycling[END_REF]. Finding the minimum Cu content to achieve this is essential to ensure that the weldability and work hardening capacity of the 5XXX alloy can be preserved.

Knowing how to choose the correct minimum level of Cu in this case requires an understanding of the microstructural changes and correlated mechanical response. To date, few studies of this kind have been performed for the low (. 1wt.%) Cu Al-Mg alloys [START_REF] Engler | Influence of copper additions on materials properties and corrosion behaviour of AlMg alloy sheet[END_REF][START_REF] Kovarik | A HRTEM study of metastable phase formation in AlMgCu alloys during artificial aging[END_REF][START_REF] Kovarik | Decomposition of an AlMgCu alloya high resolution transmission electron microscopy investigation[END_REF][START_REF] Ratchev | Precipitation hardening of an Al4.2 wt% Mg0.6 wt% Cu alloy[END_REF][START_REF] Novelo-Peralta | Characterization of precipitation in Al-Mg-Cu alloys by X-ray di↵raction peak broadening analysis[END_REF]. In contrast, many studies have been published on the related high (& 1 wt.%) Cu and low (. 2 wt.%) Mg 2XXX series Al-Cu-Mg alloys [START_REF] Ringer | Precipitation processes during the early stages of ageing in Al-Cu-Mg alloys[END_REF][START_REF] Ringer | Origins of hardening in aged Al-Cu-Mg-(Ag) alloys[END_REF][START_REF] Ringer | Response to comments on cluster hardening in an aged Al-Cu-Mg alloy[END_REF][START_REF] Reich | Origin of the initial rapid age hardening in an Al-1.7 at.% Mg-1.1 at.% Cu alloy[END_REF][START_REF] Charai | Coexistence of clusters, GPB zones, S-, S-and S-phases in an Al0.9% Cu1.4% Mg alloy[END_REF][START_REF] Starink | The origins of room temperature hardening of AlCuMg alloys[END_REF]. These precipitation hardenable alloys undergo a complex precipitation sequence,, from random solid solution to solute clusters, GPB zones, S" phase and finally S'/S phase(s) (see e.g. [START_REF] Sha | Nanostructure of aluminium alloy 2024: Segregation, clustering and precipitation processes[END_REF]. Both 2XXX-series alloys [START_REF] Marceau | Solute clustering in AlCuMg alloys during the early stages of elevated temperature ageing[END_REF][START_REF] Deschamps | In situ evaluation of the microstructure evolution during rapid hardening of an Al2.5cu1.5mg (wt.%) alloy[END_REF] and Cu modified 5XXX-series alloys [START_REF] Ratchev | Precipitation hardening of an Al4.2 wt% Mg0.6 wt% Cu alloy[END_REF] exhibit a rapid increase in yield strength in the first few seconds of aging at temperatures between 150 and 200 C. In 2XXX-series alloys this has been linked to the formation of solute clusters and GPB zones [START_REF] Marceau | Solute clustering in AlCuMg alloys during the early stages of elevated temperature ageing[END_REF][START_REF] Deschamps | In situ evaluation of the microstructure evolution during rapid hardening of an Al2.5cu1.5mg (wt.%) alloy[END_REF]. No similar experimental link has been made between precipitate phases and yield strength in the case of the low Cu/high Mg alloys [START_REF] Kovarik | Structural relationship between onedimensional crystals of GuinierPrestonBagaryatsky zones in Al-CuMg alloys[END_REF][START_REF] Ratchev | Precipitation hardening of an Al4.2 wt% Mg0.6 wt% Cu alloy[END_REF] One of the challenges in both alloy classes is our poor understanding of the structure, composition and density of solute clusters/GPB zones existing at short aging times. While the structural model proposed by Kovarik et al. [START_REF] Kovarik | Ab initio analysis of GuinierPrestonBagaryatsky zone nucleation in AlCuMg alloys[END_REF] was shown to be consistent with some of the experimentally observed characteristics of clusters/GPB zones [START_REF] Kovarik | Structural relationship between onedimensional crystals of GuinierPrestonBagaryatsky zones in Al-CuMg alloys[END_REF][START_REF] Kovarik | GPB zones and composite GPB/GPBII zones in AlCuMg alloys[END_REF], it does not predict the composition of solute clusters/GPB zones measured by atom probe tomography in model [START_REF] Marceau | Solute clustering in AlCuMg alloys during the early stages of elevated temperature ageing[END_REF],

or commercial Al-Cu-Mg alloys [START_REF] Sha | Nanostructure of aluminium alloy 2024: Segregation, clustering and precipitation processes[END_REF]. Moreover, much of the available experi-mental work on the structure and chemistry of these phases focuses on results obtained after long aging times (several hours) where characterization is notably easier [START_REF] Kovarik | A HRTEM study of metastable phase formation in AlMgCu alloys during artificial aging[END_REF][START_REF] Kovarik | Ab initio analysis of GuinierPrestonBagaryatsky zone nucleation in AlCuMg alloys[END_REF][START_REF] Kovarik | Structural relationship between onedimensional crystals of GuinierPrestonBagaryatsky zones in Al-CuMg alloys[END_REF][START_REF] Kovarik | GPB zones and composite GPB/GPBII zones in AlCuMg alloys[END_REF]. The relevance of such long term aging observations to the specific microstructure-hardening response in the paint bake cycle remains uncertain.

The aim of this work is to correlate precipitation to yield strength in Cumodified Al-3Mg alloys at times and temperatures relevant to the industrial automotive paint bake cycle, emphasizing the e↵ect of additions of < 0.5 at%Cu on the yield strength evolution. The evolution of the tensile mechanical properties and electrical resistivity are followed during aging at 160 C and 200 C for times of up to 7 hrs. To overcome the challenges of characterizing precipitates with low atomic number contrast (this limiting techniques such as STEM and SAXS), atom probe tomography has been used for characterization. A parameter free pair-correlation based technique has been used to identify the presence of clusters/particles and a model used to interpret their size and composition.

These results are subsequently fed to a precipitation hardening model to predict the experimentally measured hardening response.

Materials and Experimental Method

Two laboratory cast and hot rolled 5 mm sheets were provided for this study by the Novelis Global Technology Centre (see Table 1). This material was cold rolled to a thickness of 1 mm (80% reduction in thickness). Tensile samples were cut and solution treated at 550 C for 10 min, then quenched immediately in water. These were then artificially aged in an oil bath at 160 C or 200 C for up to 7 hrs. All uniaxial tensile tests were performed at 77 K, with an initial nominal strain rate of 10 3 s 1 . These conditions were selected so as to avoid the complicating e↵ects of dynamic strain aging [START_REF] Fazeli | Modeling the e↵ect of Al3sc precipitates on the yield stress and work hardening of an AlMgSc alloy[END_REF]. Resistivity measurements were also performed in a liquid nitrogen bath at 77K using a four-point probe with an applied current of approximately 20 mA and a reversion frequency of 30 Hz [START_REF] Raeisinia | Electrical Resistivity Measurements: A Sensitive Tool for Studying Aluminium Alloys[END_REF]. Specimens for APT were prepared by solution treating and artificially aging sheet material, following the procedure described above. Samples were cut following aging and turned into needle-shaped specimens using a conventional 2stage electrochemical polishing technique employing perchloric acid [START_REF] Miller | Atom Probe Tomography[END_REF]. Atom probe data was acquired on a Cameca LEAP 5000 XS at a base temperature of 50K, in high-voltage pulsing mode, with 20% pulse fraction, with the DC voltage adjusted to maintain a detection rate of 1 ion per 100 pulses. Data reconstruction was performed in the commercial software package IVAS 3.6.14, and using the calibration methods outlined in ref. [START_REF] Gault | Estimation of the Reconstruction Parameters for Atom Probe Tomography[END_REF][START_REF] Gault | Advances in the calibration of atom probe tomographic reconstruction[END_REF]. Precipitates were visualized by means of isocomposition surfaces [START_REF] Hellman | E cient sampling for three-dimensional atom probe microscopy data[END_REF] and composition of individual object by composition profiles derived from the regions-of-interest.

Experimental Characterization of Aging Response

The tensile response of the 0.23at%Cu alloy aged at 200 C (Figure 1) shows that its yield strength rises rapidly from 87 MPa to 150 MPa within the first 2 minutes. This is followed by a gradual but continuous increase of the yield strength over the following 18 hours (1100 min). Aside from the evolution of the yield strength, no other changes in the stress-strain response were noted. The lack of change in work hardening rate (Figure 1(b)) is particularly notable as it suggests that the hardening phase(s) are predominantly small and shearable (see e.g. [START_REF] Fazeli | Modeling the e↵ect of Al3sc precipitates on the yield stress and work hardening of an AlMgSc alloy[END_REF]). The second factor that might be expected to effect the work hardening rate (via dynamic recovery [START_REF] Kocks | Physics and phenomenology of strain hardening: the FCC case[END_REF] ) is the change of solute in solid solution. As will be shown shortly, the amount of solute removed from solid solution during aging is too small to have a noticable effect on the work hardening rate (cf. Tables 2 and3). This result mirrors the behaviour observed for all other aging conditions tested.

Figure 2 Electrical resistivity has been widely used as a mean-field characterization technique in cluster and precipitation hardening aluminum alloys [START_REF] Deschamps | Analytical Techniques for Aluminum[END_REF][START_REF] Raeisinia | Examination of precipitation in the aluminum alloy AA6111 using electrical resistivity measurements[END_REF][START_REF] Esmaeili | A study on the early-stage decomposition in the AlMgSiCu alloy AA6111 by electrical resistivity and three-dimensional atom probe[END_REF].

Changes in resistivity reflect both the loss of solute from solid solution and, in the case of su ciently fine clusters/precipitates, an increase in scattering arising from phase interfaces. While disentangling the e↵ects of solid solution and interfaces in these alloys can be challenging [START_REF] Hillel | Resistivity mechanisms during clustering in alloys[END_REF], the method gives an independent measurement for comparison against the tensile data and more local microstructural information obtained from APT or (S)TEM.

Figure 3 shows the change in electrical resistivity for those samples whose yield strength is shown in Figure 2. The higher as-quenched resistivity for the 0.23at.%Cu alloy compared to the 0.12at.%Cu alloy reflects the higher overall solute content. Consistent with the 'rapid hardening' behaviour illustrated in Figure 2, the resistivity also increases rapidly within the first 2 minutes of aging, the magnitude of the jump being much larger in the case of the 0.23at.%Cu alloy compared to the 0.12at.%Cu alloy. This rapid increase is consistent with the formation of small solute clusters that contribute significantly to conduction electron scattering (cf. results on 2XXX-series alloys [START_REF] Rosen | The aging process in aluminum alloy 2024 studied by means of eddy currents[END_REF][START_REF] Shih | Precipitation behaviors in Al-Cu-Mg and 2024 aluminum alloys[END_REF]). Following the rapid initial resistivity jump, both alloys exhibit a long period (> 100 min) of nearly constant resistivity followed by a slow decrease for the 200 C samples. To investigate the microscopic origin of the observed changes in yield strength and resistivity, atom probe tomography was performed on samples taken from the 0.23at. %Cu alloy after aging at 200 C for 20 min (typical paint bake time)

and 160 min of aging (end of the resistivity plateau in Figure 3). Ideally, one would like to compliment such local observations with other cluster/GPB zone sensitive techniques such as HR-TEM, STEM and/or SAXS/SANS to provide better statistics to these observations. As pointed out earlier, however, the lack of atomic number contrast in these alloys drastically limits the options for complimentary techniques. This has, in our opinion, been the major stumbling block for previous studies on the early stages of aging in this alloy system. Figure 4(a)

shows an APT volume measured on a sample aged for 20 min. Within this volume iso-surfaces reveal a distribution of small Mg and Cu enriched particles. The one-dimensional composition plots for the selected particles reveal the particles to contain approximately 80-90at.%Al, 10-15at.%Mg and 1-5at.%Cu.

Further aging to 160 min (Figure 4(b)) led to only minor changes in the composition and sizes of the particles based on iso-surface and one-dimensional composition plots. In both cases, the compositions were found to be similar to, but distinct from, those previously reported for clusters/GPB zones in an AA2024 alloy (4.3 wt%Cu, 1.3wt%Mg) [START_REF] Sha | Nanostructure of aluminium alloy 2024: Segregation, clustering and precipitation processes[END_REF].

Relying on conventional cluster finding algorithms has drawbacks owing to the underlying assumptions that one must make [START_REF] Gault | Atom Probe Microscopy[END_REF]. Cluster-finding methods are usually prone to parameter-selection biases [START_REF] Marquis | Applications of atom-probe tomography to the characterisation of solute behaviours[END_REF] in particular for cases where multiple morphologies of features are present [START_REF] Marceau | Quantitative atom probe analysis of nanostructure containing clusters and precipitates with multiple length scales[END_REF]. As an alternative method, a pair correlation based approach has been adopted here to interpret the results in Figure 4. This radial distribution function based method for extracting information on clusters and precipitates [START_REF] Philippe | Clustering and pair correlation function in atom probe tomography[END_REF], and its direct relationship to the formalism conventionally used in small angle scattering, has been established [START_REF] Couturier | Direct comparison of Fe-Cr unmixing characterization by atom probe tomography and small angle scattering[END_REF][START_REF] Zhao | Parameter free quantitative analysis of atom probe data by correlation functions: Application to the precipitation in Al-Zn-Mg-Cu[END_REF] and used recently for studying clustering in aluminum alloys by Ivanov et al. [START_REF] Ivanov | A combined characterization of clusters in naturally aged AlCu(Li, Mg) alloys using small-angle neutron and X-ray scattering and atom probe tomography[END_REF]. The pair correlation approach used here has the advantage of providing a self-consistent, parameter free description of the solute distribution within the dataset. This can then be interpreted to obtain information on particle size and composition when combined with a suitable model.

We use a modified definition of the pair correlation function (PCF) compared to the one introduced in [START_REF] Geuser | 3d atom probe study of solute atoms clustering during natural ageing and pre-ageing of an Al-Mg-Si alloy[END_REF]. The form used here allows the PCF to be expressed in an equivalent way to the form used in small angle scattering in the case of an isotropic two-phase system where the composition is uniform within the two phases [START_REF] Debye | Scattering by an Inhomogeneous Solid[END_REF][START_REF]Small angle x-ray scattering[END_REF][START_REF] Couturier | Direct comparison of Fe-Cr unmixing characterization by atom probe tomography and small angle scattering[END_REF][START_REF] Ivanov | A combined characterization of clusters in naturally aged AlCu(Li, Mg) alloys using small-angle neutron and X-ray scattering and atom probe tomography[END_REF]. We define a pair correlation function as,

i j (r) = c i 0 c i j (r) c i 0 c j 0 (1)
where c j 0 is the average concentration of species j in the alloy and c i j (r) is the concentration of species j at a distance r from an atom of species i averaged over all atoms of species i. i j (r) is the correlation between the concentration fluctuations in element i and element j, i.e. ⌦ c i c j ↵ . Equation 1 can be written in terms of two contributions,

i j (r) = ij (0) 0 (r) (2)
The function 0 (r) is a normalized correlation function, such that 0 (0) = 1 and 0 (r ! 1) = 0. The meaning of 0 (r) is particularly intuitive in the case of a single object in a (infinitely) large homogeneous volume. In this case 0 (r) is the normalized autocorrelation; it is the intersection volume of the object and its 'ghost' displaced by a distance r normalized by the object volume. For simple shapes, this function can be determined analytically. For example, for spherical particles of radius R, 0 (r) is [START_REF] Guinier | Small-angle scattering of X-rays, Structure of matter[END_REF],

sphere 0

(r) =

> <

> :

1 3r 4R + r 3 16R 3 (r  2R) 0 (r > 2R) (3)
In the case of randomly distributed precipitates whose radii are distributed (e.g. following a log-normal distribution) a volume weighted numerical integration of Equation 3 can be performed to obtain 0 (r).

It is important to note that 0 (r) is a unique function of the size and geometry of the second phase. The e↵ect of matrix/particle composition enters entirely through i j (0) which is a measure of the compositional contrast between the phases. In the case of the correlation between atoms of the same type,

i i (0) = f v (1 f v ) c i p c i m 2 (4)
while in the case of the correlation between two di↵erent types of atoms we obtain,

i j (0) = f v (1 f v ) c i p c i m c j p c j m (5)
In the above equations, c p and c m refer to the particle and matrix compositions, c 0 to the average compositions and f v to the volume fraction of particles.

It is important to note that for a pseudo-binary situation, i.e. if only one type of objects are present, the 3 PCFs i i (r) , j j (r) and i j (r) should all be proportional, the factor of proportionality being related to the fraction of objects and the composition contrast. Moreover, based on the definitions given above,

i j (0) = q i i (0) j j (0) (6)
In the present case, we have calculated the Mg-Mg, Cu-Cu and Mg-Cu pair correlations (Figure 5) from the APT data shown in The results in Figure 5 can be interpreted in terms of particle/matrix compositions and particle sizes if a model for the phases is proposed. Here, it has been assumed that the particles in the APT volumes are spherical, the radii of the particles following a log-normal size distribution. Analysis of the iso-contour data in Figure 5 suggests that the aspect ratio of the particles increases on aging from ⇠1 at 20 min to close to 2 at 160 min of aging. We have, nevertheless, retained the assumption of spherical particles so as to keep the number of fitting parameters as small as possible. Moreover, the e↵ect of this approximation on the prediction of yield strength based on particle size presented below, is expected to be nearly insensitive to this assumption for such small aspect ratios [START_REF] Nie | Microstructural design of high-strength aluminum alloys[END_REF].

To account for the fact that the pair correlations in Figure 5 do not scale with one another, we have considered that two populations of particles exist, one being richer in Cu with a larger correlation length compared to the other.

From fitting Mg Mg (0), Cu Cu (0) and Cu Mg (0) one would like to be able to independently obtain the particle compositions and volume fractions (for the assumed two particle types) as well as the matrix composition. In order to do this one extra piece of information needs to be supplied. In the present case we have assumed that the particles contained 80 at% Al based on the profiles in Figure 4.

Under the above assumptions it was possible to obtain two size distributions and particle compositions from the APT volumes (Figure 6). The resulting fits to the pair correlations are shown as solid, coloured, lines superimposed on the experimentally determined pair correlations (symbols) in Figure 5. The values obtained from the fit are also given in Table 2. Consistent with the composition profiles shown in Figure 4, the particles are seen to be systematically richer in Mg than Cu, with the Cu:Mg ratio being highest in the largest precipitates.

With increased aging time, both distributions are seen to shift to larger particle sizes, again qualitatively consistent with the results from the isocomposition plots (Figure 4). Interestingly, the volume fractions of the two populations are observed to remain constant, suggesting a process of coarsening rather than growth. This interpretation would also be consistent with our resistivity results (Figure 3) when interpreted using resistivity models [START_REF] Osamura | Resistivity maximum during Guinier-Preston zone formation in an Al-4 wt% Cu alloy[END_REF][START_REF] Hirosawa | Comparison between Resistivity Changes and Monte Carlo Simulation for GP Zone Formation in Al–Cu Base Ternary Alloys[END_REF] that predict cluster contributions to the resistivity proportional to volume fraction, and provide also an explanation for the resistivity decrease at long aging times observed at 200 C, likely due to the coarsening of the particle distribution. Table 2: The mean particle size (hRi) and standard deviation (S) of assumed log-normal particle size distribution, volume fraction (fv), number density and composition of particles obtained by fitting to the data in Figure 5 assuming two log-normal particle size distributions in each aging condition. The Al content was fixed at 80 at% based on the composition profiles extracted in Figure 4.

Predicting Yield Strength Evolution from Microstructural Observations

If the features seen in APT (Figures 4) are responsible for the evolution in yield strength reported in Figure 2 it should be possible to use the data from Figure 6 to predict the evolution of yield strength with time. Calculating the strength of such a distribution of precipitates must be done carefully as the way in which one chooses to calculate the average strength of such an assembly can have a strong influence on the prediction [START_REF] Fazeli | Modeling the e↵ect of Al3sc precipitates on the yield stress and work hardening of an AlMgSc alloy[END_REF][START_REF] Vaucorbeil | The E↵ect of Obstacle Strength Distribution on the Critical Resolved Shear Stress of Engineering Alloys[END_REF]. Here, for simplicity, We take a slightly modified approach to the one originally proposed by Deschamps et al. [START_REF] Deschamps | Influence of predeformation and ageing of an AlZnMg alloyII. Modeling of precipitation kinetics and yield stress[END_REF] where the average strength of a distribution of obstacles is calculated as the average of the individual obstacle strengths [START_REF] Fazeli | Modeling the e↵ect of Al3sc precipitates on the yield stress and work hardening of an AlMgSc alloy[END_REF]. It has been shown, by comparing to the results of areal glide simulations, that this approach provides a lower limit estimate to the strength of a distribution of obstacles, the predictions being better for narrower and weaker obstacle populations [START_REF] Vaucorbeil | The E↵ect of Obstacle Strength Distribution on the Critical Resolved Shear Stress of Engineering Alloys[END_REF].

We start by assuming that the strength of an individual particle is proportional to the radius of the circle (r) formed by the intersection of the spherical particle and the glide plane [START_REF] Marceau | Analysis of strengthening in AA6111 during the early stages of aging: Atom probe tomography and yield stress modelling[END_REF]. The distribution of circle radii (f (r)) formed by cutting the spherical particles (having a distribution of radii g (R), where R is the sphere radius) can be calculated from Wicksell's fundamental integral equation [START_REF] Marceau | Analysis of strengthening in AA6111 during the early stages of aging: Atom probe tomography and yield stress modelling[END_REF][START_REF] Baddeley | Stereology for Statisticians[END_REF],

f (r) = r R Z 1 0 g (R) dR p R 2 r 2 (7)
where R is the mean value of the sphere radii.

The contribution of the precipitates to the macroscopic yield stress is defined as,

ppt = M Gb Ls ⌧ ⇤ ( 8
)
where M is the Taylor factor, G is the shear modulus of aluminum and b is the magnitude of the Burgers vector. The average square spacing of precipitates in the glide plane, Ls , is taken following Ardell [START_REF] Ardell | Precipitation hardening[END_REF] as,

Ls = ✓ 2⇡ 3f ◆ 1/2 R ( 9
)
The normalized average strength of the ensemble of particles, ⌧ ⇤ is calculated as,

⌧ ⇤ = Z 1 0 ⌧ (r) ⇤ f (r) dr (10)
In this expression, ⌧ ⇤ is the strength of a randomly distributed population of particles whose size in the glide plane is r. In the approach originally proposed by [START_REF] Deschamps | Influence of predeformation and ageing of an AlZnMg alloyII. Modeling of precipitation kinetics and yield stress[END_REF] the strength of obstacles was assumed to obey Friedel's law, this only being strictly valid in the limit of weak obstacles. This approach was generalized by [START_REF] Fazeli | Modeling the e↵ect of Al3sc precipitates on the yield stress and work hardening of an AlMgSc alloy[END_REF] to include obstacles of all strengths by using the classic result of Foreman and Makin [START_REF] Brown | Dislocationparticle interactions[END_REF] Here, we take the same approach, but use an updated relationship between obstacle strength and particle size arising from computer simulations [START_REF] Vaucorbeil | Dislocation glide through non-randomly distributed point obstacles[END_REF],

⌧ (r) ⇤ = 8 > < > : 0.9 ⇣ r rc ⌘ 3/2 ✓ 1 1 6 ⇣ r rc ⌘ 5 ◆ if r < r c 0.75 if r r c (11)
The parameter r c in this expression is the critical size at which the particles transition from being shearable to non-shearable. It was noted, in relation to Figure 1, that the lack of change in work hardening indicates the presence of shearable particles. When considering a distribution of particle sizes one must, however, consider the possibility of sizes that span a range of sizes from below to above r c .

The presence of two particle distributions (one lower in Cu, the other higher in Cu) leads to a question of whether one should use di↵erent r c for each distribution. It has been argued, though not proved definitively, that precipitation/cluster hardening in Al-Cu-Mg alloys is sensitive to the composition of the phases present [START_REF] Marceau | Solute clustering in AlCuMg alloys during the early stages of elevated temperature ageing[END_REF]. Here, we have chosen to fix a single value of r c so as to minimize the number of adjustable parameters. Even with a single value of r c , the two particle distributions (Figure 6) lead to two values of ppt according to Equation 8. Considering that the strengths and densities of these two particle distributions will be similar, the most appropriate method for their addition is by (see e.g. [START_REF] Vaucorbeil | The superposition of strengthening contributions in engineering alloys[END_REF]).

ppt = q 2 ppt,1 + 2 ppt,2 (12)
Finally, in order to calculate the total yield strength, we also have to factor in the solid solution contribution. From the particle compositions and volume fractions, we can calculate the remaining Cu and Mg in solid solution based on the bulk composition of the alloy. As in the case of the precipitate contribution to the yield strength, the net contribution from the two solid solution alloys is expected to follow [START_REF] Vaucorbeil | The superposition of strengthening contributions in engineering alloys[END_REF][START_REF] Vaucorbeil | The E↵ect of Obstacle Strength Distribution on the Critical Resolved Shear Stress of Engineering Alloys[END_REF].

ss = q 2 ss Cu + 2 ss Mg = r ⇣ k Cu X 2/3 Cu ⌘ 2 + ⇣ k Mg X 2/3 Mg ⌘ 2 (13)
where the solid solution hardening coe cients, k Cu = 81.8 MPa/at.% 2/3 and k Mg =27MPa/at.% 2/3 are taken from [START_REF] Myhr | Modelling of the age hardening behaviour of AlMgSi alloys[END_REF].

Table 3 provides from precipitates and solid solution to the overall yield strength. The relative average precipitate strength is also reported where = F m /2 where is the dislocation line tension and = 1 corresponds to the Orowan strength (r = r c ). To compute the total yield strength, the precipitate and solid solution contributions are summed as in Equation 13and 12 owing to their similar magnitudes. Finally, a constant term, 0 , has been added to account for the e↵ect of grain size,

ys = 0 + q 2 ss + 2 ppt ( 14
)
The value of 0 was obtained directly by measuring the yield strength of a fully solution treated sample and subtracting the solid solution contribution calculated from Equation 13.

Using the above methodology with the particle size distributions from Figure 6 there remains only one unknown parameter(r c ) needed to calculate the yield stress. In order to fit the yield strength measured after 20 min of aging at 200 C (170 MPa), a value of r c = 3.95 nm needs to be used. This is very close to the equivalent value (r c = 5 nm) used by Deschamps et al.

precipitation hardening response of an Al-1.1at%Cu-1.7at%Mg alloy [START_REF] Deschamps | In situ evaluation of the microstructure evolution during rapid hardening of an Al2.5cu1.5mg (wt.%) alloy[END_REF]. As noted in the introduction, the qualitative similarity of the hardening response of the low Cu alloys studied here and more conventional 2XXX series alloys has been interpreted as evidence of similar microstructural evolution on aging. Comparison of the observations presented here to those from more conventional Al-Cu-Mg alloys [START_REF] Marceau | Solute clustering in AlCuMg alloys during the early stages of elevated temperature ageing[END_REF][START_REF] Sha | Nanostructure of aluminium alloy 2024: Segregation, clustering and precipitation processes[END_REF] suggests similarities and di↵erences. In the APT observations reported in [START_REF] Marceau | Solute clustering in AlCuMg alloys during the early stages of elevated temperature ageing[END_REF] and [START_REF] Sha | Nanostructure of aluminium alloy 2024: Segregation, clustering and precipitation processes[END_REF], following aging similar to that used here, the dominant feature of the microstructure was reported as solute clusters. Such clusters were described as roughly spherical solute enriched regions containing on average 24 solute atoms [START_REF] Sha | Nanostructure of aluminium alloy 2024: Segregation, clustering and precipitation processes[END_REF]. Upon aging up to peak strength (80 hours at 170 C) three di↵erent features were found to dominate; solute clusters (up to 100 atoms in size, < 4 nm in diameter), rod-like GPB zones (< 4nm in diameter and 10-60 nm in length) and large plate like S-phase precipitates.

The features seen in Figure 4 and quantified via the pair correlation analysis in Figure 6 seem to situate somewhere between the features reported at short aging times and long aging times in the Al-Cu-Mg alloys noted above [16, The solute clusters reported in [START_REF] Sha | Nanostructure of aluminium alloy 2024: Segregation, clustering and precipitation processes[END_REF] after 30 min of aging at 170 C are similar in size and morphology to the average size of the Cu lean particles in Figure 6 after 20 min of aging. At the longer aging times, the size of the Cu rich particles along with the tendency for more rod-like morphologies suggests a transition towards more GPB like particles, according to their definition in [START_REF] Marceau | Solute clustering in AlCuMg alloys during the early stages of elevated temperature ageing[END_REF].

A significant di↵erence between the clusters/GPB zones reported in [START_REF] Sha | Nanostructure of aluminium alloy 2024: Segregation, clustering and precipitation processes[END_REF] and the particles observed here are the particle chemistries. While the cluster/GPB zones in [START_REF] Sha | Nanostructure of aluminium alloy 2024: Segregation, clustering and precipitation processes[END_REF] were found to have Mg:Cu ratios on the order of 1.1-1.3, here the Mg:Cu ratio was found to range from as high as 16 (in the Cu lean particles at 20 min of aging) to as low as 2 (in the Cu rich particles after 160 min of aging).

What is consistent with regard to chemistry across both studies, however, is that in all cases the particles are predominantly made up of Al; ⇠ 80 at% in this study and between ⇠ 85 and ⇠ 90at% in [START_REF] Sha | Nanostructure of aluminium alloy 2024: Segregation, clustering and precipitation processes[END_REF]. The enrichment in Mg found here may not be entirely surprising considering the much higher bulk Mg:Cu ratio (14:1) for the alloy studied here compared to 0.97:1 in the alloy studied in [START_REF] Sha | Nanostructure of aluminium alloy 2024: Segregation, clustering and precipitation processes[END_REF]. This enrichment, combined with the high Al content of the particles, is a significant factor in the relatively high number density of particles (N v ⇡ 10 23 m 3 ) despite the low Cu levels employed. It is interesting to note that the behaviour seen here where small amounts of Cu can catalyze precipitation in Al-Mg alloys qualitatively mirrors the behaviour seen in Al-Cu alloys where rapid hardening is only observed when small additions of Mg are made to the alloy [START_REF] Marceau | Solute clustering in AlCuMg alloys during the early stages of elevated temperature ageing[END_REF]. Indeed, the precipitation strengthening observed here is quite comparable to that reported in more conventional Al-Cu-Mg alloys despite a lower number density of particles. For example, an Al-2.5wt%Cu-1.5wt%Mg alloy has been reported to harden by ⇡ 80 MPa after 100-200 min of aging at 200 C [START_REF] Deschamps | In situ evaluation of the microstructure evolution during rapid hardening of an Al2.5cu1.5mg (wt.%) alloy[END_REF].

In the case of the 0.23at%Cu alloy studied here, = 108 MPa evaluated after 160 min of aging at the same temperature. The strengthening potential of these clusters/GPB zones may relate to a point made in relation to strengthening in Al-Cu-Mg alloys [START_REF] Marceau | Solute clustering in AlCuMg alloys during the early stages of elevated temperature ageing[END_REF] where it was argued that particles richer in Mg seemed to be more e↵ective at hardening compared to those rich in Cu.

The ability even very small Cu additions to e ciently catalyze the formation of a relatively high number density of predominantly Al-Mg strengthening particles at short aging times is emphasized by the results obtained for the alloy containing only 0.12at%Cu (Figure 2). One can double the solutionized yield strength of both alloys studied here ( = 90 MPa for 0.23at.%Cu and = 67 MPa for 0.12at.%Cu) by aging for 20 min at 200 C, this despite the fact that one alloy has half as much Cu as the other. These results support those originally reported in [START_REF] Court | The Ageing and Thermal Recovery Behaviour of Al-Mg-Cu Alloys[END_REF] where strengthening was achieved for alloys containing as little 0.08at%Cu. This is important given the aim of counterbalancing the ⇠ 25% loss in strength during the paint bake cycle [START_REF] Court | The Ageing and Thermal Recovery Behaviour of Al-Mg-Cu Alloys[END_REF] with the minimal addition of Cu.

Finally, we point to the significant potential for further studies focused on better understanding the precipitation sequence and kinetics in these alloys. As noted above, the nearly identical aging response at 160 C and 200 C is surprising. This observation suggests that there may be competing mechanisms leading to an optimal strengthening in this temperature range. Indeed, tests performed at lower temperature (100 C, not shown here) show lower strength for the same aging time. The competition between the kinetics of excess vacancy annihilation and vacancy trapping by solute clusters may be a fruitful area for further exploration in regards to this phenomenon [REF]. The observed linear dependence of yield strength on logarithmic time also warrants further study. As noted above, this behaviour is not unique to the work reported here but has also been reported by both Court and Lloyd [START_REF] Court | The Ageing and Thermal Recovery Behaviour of Al-Mg-Cu Alloys[END_REF] and Ratchev et al. [START_REF] Ratchev | Artificial Ageing of Al-Mg-Cu Alloys[END_REF]. It was suggested that the lack of variation in volume fraction of clusters/GPB zones on aging at 200 C is most coherent with an interpretation of particle coarsening. Further to this point, if one were to adopt a simple model for coarsening kinetics see e.g. [START_REF] Shercli↵ | A process model for age hardening of aluminium alloysI. The model[END_REF] then one would expect an aging response that was nearly linear in log time and 20 whose slope would be independent of aging temperature and composition. To be able to confidently confirm or deny this interpretation would require many further studies on other alloys and other aging temperatures.

Summary

It has been shown

alloy can lead to significant precipitation hardening, the observed precipitates being similar in size and chemistry to clusters/GPB zones previously reported in more conventional 2XXX-series alloys and model Al-Cu-Mg alloys. Using data extracted directly from APT measurements, the size distribution of particles could be used to predict the yield strength after aging 160 min at 200 C following calibration of the shearable/non-shearable radius from data collected after 20 min of aging. While the strengthening particles share similar size and geometry to clusters/GPB zones in higher Cu/lower Mg alloys [START_REF] Sha | Nanostructure of aluminium alloy 2024: Segregation, clustering and precipitation processes[END_REF], the chemistry of the particles observed here were much leaner in Cu. This observation helps explain the relatively large number density of clusters/GPB zones despite the low bulk Cu content. Given the appreciable strengthening observed in these low Cu alloys, precipitation hardening in alloys containing very low Cu contents should be able to readily compensate for recovery induced softening in the industrial paint bake cycle. The question of the e↵ectiveness of precipitation hardening in pre-deformed alloys, however, is the subject of ongoing work.

Figure 1 :

 1 Figure 1: a) The tensile true stress-true strain response for the alloy containing 0.23at.%Cu in the solid solution state (0 min of aging), after 2 min, 20 min, 160 min and 420 min aging at 200 C. b) The same data as in (a) but plotted as a Kocks-Mecking plot showing the collapse of the work hardening rates when the yield strength is subtracted from the flow stress.

Figure 2 :

 2 Figure 2: The evolution of the 0.2% o↵set yield strength for the 0.23at.%Cu alloy aged at 160 and 200 C and the 0.12at.%Cu alloy at 200 C for times ranging from 2 min to 1100 min. The yield strength of the solution treated sample is shown on the far left side of the plot. Lines are drawn as a guide to the eye.

Figure 3 :Figure 2 .

 32 Figure 3: The time evolution of electrical resistivity for the 0.23at%Cu alloy at 200 C and 160 C and the 0.12at.%Cu alloy aged at 200 C for the same aging times used to produce Figure 2. The resistivity of the solution treated samples is shown on the far left. Error, shown as the bars on each data point, was calculated based on the uncertainty in measured resistance and area. The lines are shown as a guide to the eye.

Figure 4 :

 4 Figure 4: Atom probe volumes measured on samples aged for a) 20 min and b) 160 min at 200 C. In each case, iso-surfaces corresponding to a) 3.13 atoms/nm 3 Mg and 0.5 atoms/nm 3 , and b) 3.5 atoms/nm 3 Mg 1.0 atoms/nm 3 Cu are plotted to reveal the presence of small Mg and Cu rich particles. One dimensional composition plots are provided along two perpendicular directions, the results being representative of the observations made in the other particles.

Figure 4 .

 4 If the particles observed in Figure4belonged to a single population with a uniform composition, then all three pair correlation functions should be proportional to one another. Careful observation of the Mg-Mg and Cu-Cu pair correlations shows that this can't be true as the correlation length for the Cu-Cu pair correlations is noticeably longer than that for the Mg-Mg pair correlations.

Figure 5 :

 5 Figure 5: Experimental (symbols) and fit (coloured lines) pair correlation functions for a) Cu-Cu pairs, b) Mg-Mg pairs and c) Mg-Cu pairs from the APT datasets shown in Figure 4. If all particles belonged to the same population having a uniform composition, then the three figures should simply scale with one another.

Figure 6 :

 6 Figure 6: The best fit log-normal size distributions corresponding to the pair correlations shown in Figure 5. Left: aging for 20 min at 200 C. Right: aging for 160 min at 200 C. The green and red areas account for the respective amounts of Cu and Mg in the particles. The results are given as a volume weighted distribution, or dfv dR .

  and A. Sturm are acknowledged for their support and the running of the APT and FIB facility at MPIE. H. Zhao acknowledges the Chinese Scholarship Council for her PhD scholarship. S. Medrano acknowledges the Mexican National Council of Science and Technology (CONACyT) for his PhD scholarship. C. W. Sinclair and S. Medrano acknowledge the Natural Sciences and Engineering Research Council of Canada for financial support. Finally Novellis Inc. is thanked for the provision of materials.

Table 1 :

 1 Composition obtained by OES of alloys studied. Balance is aluminum.

	Mg	Cu	Fe	Si	Ti	Ni	Mn	Zn	Cr
	0.23 at.%Cu Alloy (at.%) 3.23 0.229 0.048 0.048 0.008 0.003 0.0005 0.00008 0.0005
	(wt.%) 2.90 0.54	0.1	0.05 0.015 0.007 0.001	0.002	0.001
	0.12 at.%Cu Alloy (at.%) 3.20 0.115 0.046 0.046 0.007 0.003 0.0005 0.0012 0.0005
	(wt.%) 2.96 0.28	0.1	0.05 0.014 0.007 0.001	0.003	0.001

Table 3 :

 3 Parameters used/calculated in predicting the yield strength of the alloy after aging for 20 min and 160 min at 200 C. The relative strengths of the particles ( = Fm/2 ) are also reportedHaving established a value for r c one can test the method's ability to predict the yield stress after aging of 160 min, the results being shown in Table3. One can see that our predicted value of the yield stress (195 MPa) is nearly identical

	Aging Time	0 (MPa)	ss Cu (MPa)	ss Mg (MPa) rc (nm)	ppt (MPa)		ppt (MPa)		ys (MPa)	ys (MPa)
	at 200 C					Rich) (Cu Rich) (Cu Lean) (Cu Lean) Predicted Experimental
	20 min	20	30	58	3.95	104	0.5	87	0.2	170	170
	160 min	20	29	58	3.95	106	0.7	124	0.5	195	196
	to that measured experimentally (196 MPa). It is important to note that this
	good prediction would not have been possible if, in Equation 8 we would have
	used the mean radii of the distributions to calculate ⌧ ⇤ [21]. Indeed, fitting
	the yield stress after 20 min of aging would require r c = 3.8 nm leading to a
	predicted yield stress of 211 MPa in contrast to the experimental value of 196
	MPa.