Shoyeb Al

Mamun Shohag

Cloud Computing and the Business Consequences of ERP Use

Keywords: Cloud, ERP, Business Consequence

A Cloud Computing system is intended to improve and automate the controlling single point operations. By using a single point of control, this goal is accomplished through the elimination of duplicate entry and the contribution of data integrity, detailed drilldown, simple training, manageable support, minimal IT maintenance, easy upgrades and reduced costs. Overall, the advantages of cloud computing usage fulfill the original intentions of business as it allows process manufacturers to manage their business as simply and efficiently as possible. Enterprise Resource Planning (ERP) software is designed to improve and auto-mate business processes operations. However, there are many unnecessary administrative, procedural costs and delays often associated with this practice. Examples include duplicate data entry, data corruption, increased training, complicated supplier relations, greater IT support and software incompatibilities. Purpose of this system is Single Point of Control, Duplicate Entry Elimination, Data Integrity, Detail Drill Down, Basic Training, Manage Support, Security, Minimal IT Maintenance, Easy Upgrades, and Reduce Costs etc.

INTRODUCTION

Now every organization are in a global world where all the business are very much familiar to use technology for prosper, prospect as well as daily work. Business has willing to do all the works in respect of saving money, time and energy and also for the fortification of their assets and belongings in an effective and efficient way.

ERP software is very much dependent on Software platform, Database, Intelligence, security and other third party software. Some time it is not possible for single software development firm to treat every dependency and feature in a same manner [START_REF] Park | A Conceptual Model of ERP for Small and Medium-Size Companies Based on UML[END_REF].

Cloud computing can ensure the benefit of both vendor and the business user. This is only part of the challenge, though. Forecasted growth will generally have at least some impact on IT operating costs. Here assumed that cloud is not only created for the ERP the total support and establishment of ERP will be from a cloud [START_REF] Shohag | Deploying an Application on the Cloud[END_REF].

Organizations have goals to implement this proposed system let the business organization define their business process, no dependency on predefined ERP business flow. To support and develop a cost effective and user friendly Cloud based ERP system providing high quality and robust system dependency. There is a multiple venture idea to support and maintain the system. This system can offer reliable and paramount service to the IT management for business organizations [START_REF] Sudha | Investigation on Efficient Management of workflows in cloud computing Environment[END_REF].

The rest part of this paper have been structured as, literature review has been point out in section 2. Section 3 holds the traditional ERP with business cases. Our proposed cloud computing system has been placed in section 4. A comparison among present ERP flowchart and proposed ERP flowchart has been given in section 5. Discussion and analysis part is in section 6. A conclusion has been made in section 7.

LITERATURE REVIEW

The authors in [START_REF] Shohag | Deploying an Application on the Cloud[END_REF] mainly focused on cloud, cloud computing and its myriad application. They describe various cloud computing paradigm and types of clouds those includes public cloud, private cloud, community cloud and hybrid cloud. A cloud computing architecture has been describes with existing features. Finally, they have deployed a cloud computing system and showed the feature that differentiates with the existing cloud computing process.

In [START_REF] Bala | Intensification of educational cloud computing and crisis of data security in public cloud[END_REF], the author works with cloud computing for educational system and more specifically on the security issues of public clouds. At the end, he has provided a guide line to develop a cloud computing for education system. Finally the author suggest few issues to develop a educational cloud computing system those includes working principles, significance of development, growth of the system and implications of the education.

An investigation on cloud computing has been done in [START_REF] Sudha | Investigation on Efficient Management of workflows in cloud computing Environment[END_REF]. The authors tested the performance of cloud computing systems running in a virtual environment via remote, and wide-area network resource access. Their test result showed that virtual environment system having short job runtime of a work flow has the good compute time scale that can be effected by resource scheduling delays and wide area communications.

According to [START_REF] Ahmad | Cloud Computing: A solution to Geographical Information Systems (GIS)[END_REF], the authors' works on cloud computing system for GIS (Geographical Information System) and they have made a large evaluation on cloud computing approach to GIS. Finally, a multi-tiered architecture for GIS Cloud System has been proposed that holds mainly two components those are GIS cloud

Methodology

web interface and GIS server. The GIS server has been divided into five sub layer to get the best performance from the shared resource like databases, configuration, server logic, server side utilities, communication interfaces and high powered processing infrastructure.

Importance and advantage of cloud computing on web application have been showed in [START_REF] Liladhar | Implementation of Cloud Computing on Web Application[END_REF]. The author's remarks on few issues related to cloud computing for web application like price, simplicity, flexibility, collaboration, privacy and security. They also discussed how power loss occurred with the component of computer and they think that distributing highvoltage DC power throughout the data centre may reduce the loss.

The authors in [START_REF] Mukherjee | Cloud Computing: Future Framework for e-Governance[END_REF] focused on frame work of future egovernment for cloud computing. They observed that present egovernment frame work is not able to map all users' criteria. They have been proposed an new effective e-government frame work which is more intelligent and able to filled up all user criteria's.

An algorithmic approach and eco-friendly cloud computing have been implemented and tested in [START_REF] Mukherjee | Green Cloud: An Algorithmic Approach[END_REF]. They also works on green cloud computing. The green cloud empathises on the power consumption by the cloud computing approaches. They have remarks on proper security management system for their proposed of cloud computing approaches for future endeavours.

A mathematical model for cloud computing has presented in [START_REF] Mukherjee | Development of Mathematical Model for Market-Oriented Cloud Computing[END_REF].

Where the authors implemented a mathematical model based on marketing oriented cloud computing. An algorithm also been implemented from the mathematical model. They have been tested their proposed system setting up a private cloud based on Ubuntus 10.04 Server edition. Their test result shows that as stochastic demand gradually increases, the proposed objective function slowly increases and provides more optimized value when the stochastic demand is within a moderate to maximum range although they didn't find best performance when maximum stochastic demand held's.

According to [START_REF] Armbrust | Above the Clouds: A Berkeley View of Cloud Computing[END_REF], the authors focused on Software as a service providers (cloud user) and cloud providers. They have defined cloud, is the combination of data centre hardware and software. They also define the public cloud as, when a cloud have prepared with the pay-as-you go manner for public and these services are sold called utility computing. They have predict cloud computing will grow fast and developer should account on how smoothly a single virtual Machine serve the services. In addition, they also remarks three points those includes scale down and up of application software, upgrading of infrastructure software and hardware system (putting idle portions of the memory, disk, and network into low power mode, processors should work well with VMs, flash memory should be added to the memory hierarchy, and LAN switches and WAN routers must improve in bandwidth and cost).

In [START_REF] Hayes | Cloud computing[END_REF], author works on cloud computing because of the migration of software system to distance server that is accessible remotely. They have shown the evolution of cloud computing and predicts the future of cloud computing is less than clear, showing few example that treated as directions. They have mention about WordStar for the web that first attracted people to cloud computing. They also mentioned about Google docks as earlier developed application for WordStar. In enterprise cloud computing, the first was Salesforce.com that was founded in 1999. They also focused on cloud infrastructure and cloud OS.

The author in [START_REF] Mladen | Cloud Computing -Issues, Research and Implementations[END_REF], focused on the concept of cloud computing, addressed few issues related to research topics and a cloud implementation that works with VLC technology. The author shown his experience with VLC technology is excellent and he is working on additional functionalities and features for future improvement of cloud frame work of cloud computing.

The authors in [START_REF] Armbrust | Above the Clouds: A Berkeley View of Cloud Computing[END_REF], initially defined the cloud computing and they have remarks three aspect from hardware point of view like, the appearance of infinite computing resources available on demand, the elimination of an up-front commitment by cloud users and the ability to pay for use of computing resources on a short-term basis as needed. They have discussed about the cloud computing economies and showed the benefits of cloud computing over conventional data centre. They also finds out top ten obstacles and opportunities for growth of cloud computing.

BUSINESS CASE

To make expenses predictable, and cloud computing-based solutions provide an opportunity to do so. Centralized ERP and other venture to support the system will reduce the high cost for the Business organization. It will ensure the performance, reliability, robustness which is very necessary for a business organization. Various ventures will introduce to maintain and support the system. Identical performance will be provided to various business organization and centralized data base for everyone.

Infrastructure is expensive. Without the scale available to larger businesses, in fact, it's disproportionately costly. Even when there aren't unexpected outages, basic maintenance, and growth considerations can lead to necessary increases in IT budget, consuming capital that could invest in other projects instead.

The opening up of Small and Medium Enterprises (SMEs) market for ERP. The high IT infrastructure and ERP Implementation project costs has acted as entry barrier. The "Cloud Computing" model will offer very economical way to embrace and implement ERP for these companies.

ERP Methodology Fig 1: Traditional ERP

Traditional ERP has a methodology that consists of four components. The software component is the expression of the ERP. The process flow is defining the flow of information among the ERP. Customer mindset set the work flow of an organization. This work flow set up will be done by compromising in between the before implementation way of work and after implementation way of work. Lastly, change management needed different state of ERP i.e. customer attitude resistance and any kind of business process changes. Proposed system has some distinctive change that explained in the next section [START_REF]Project Management in Enterprise Resource Planning (ERP) Implementation[END_REF].

PROPOSED SYSTEM FEATURES

Defining the business process will be on the safe side of business organization. Business organization will follow a standard business process, which can be defined by ERP system. Our proposed system for Cloud computing has been placed in Fig 2 bellows.

Definition and connectivity of database system will be provided by vendor. Business organization has to have an option to choice database depending on their budget and intelligence they need. Database definition and specification is defined such a way that different business organization has an access to their own database. Reporting tools will be general for all business organization. Business organization can define their simple reports by their own skill. If the report is complex, business organization will consider vendor to provide them support. Report integration (RI) will be provided by the vendor to get reliable and specific report for the organization.

Security is a vast issue for ERP. Vendors will provide security to their cloud, application and database separately. Security and encryption may be provided by the different vendor to make a reliable system. Some time ERP need integrated hardware mostly to read Infrared tag, RFID (Radio Frequency Identification) tag etc. These integrated hardware driver software will be supported by the cloud system.

To implement most sophisticated ERP system there is a vast dependency on a reliable Operating system. Vendor will chose an operating system to be on cloud and give an elevated performance to the Business organization. This operating system should be easily configurable and maintainable by vendors. Operating system is also very important to support third party software.

Cloud server should be robust and reliable in its performance. IT service of different organization will be provided in a single point. As Cloud Computing becomes more mature, there will be opportunity for ERP Service Vendors on two fronts. Existing big companies will start migrating to "Cloud" to reduce capital and operational IT expenditure.

All IT support, Maintenance and training provided by the vendor end. Traditional systems are very high at cost in these issues. Cost will be nominal. Every R&D dependent on the cloud environment will be pointed on the single point service.

The ERP cloud will be access by Administrator, User and Vendors. Vendor will create and support the cloud to give its optimum performance to the Business user. Version controlling and upgrades will centrally provided by the vendors. Administrators from both end -business organization and vendor will define the business process, check the performance, reliability and robustness of ERP. User of the Cloud ERP system will make an efficient use of the system. This end user will authenticate and corroborate that the system is fit for their purpose.

ERP IMPLEMENTATION TECHNIQUE

Traditional ERP implementation plan has been studied and point out with the sub section. A cloud based ERP implementation plan has designed and showed with the sub section of this part that gives us a clear understanding and differentiates from each other.

Traditional ERP Implementation Plan

According to [START_REF] Dr | llau@longwood.edu, developing a successful implementation plan for ERP[END_REF], the flow chart of traditional ERP system has mentioned bellows in the Fig 3.

Proposed Cloud ERP Implementation Plan

ACKNOWLEDGEMENT

First, our thanks to Almighty "ALLAH" to keep us safe and sound for the time being. We are also thankful to our parents for their unbounded love and blessing for ever and friends for their great cooperation.

CONCLUSION

Our aim was to design a cloud based ERP business consequence that has been done and discussed in the previous sections. We have also showed how our proposed cloud based ERP system would be easy of use and cost effective comparing with present ERP flowchart. The weakness of our proposed ERP model has also been discussed .Considering the cost effectiveness and ease of use, our proposed system has a better prospect in the filed of ERP based automatic business system. The Cloud Computing paradigm is yet to prove itself as a reliable and mature IT services model which is not only promises to cut the IT infrastructure costs but will also provide businesses to reap benefits from faster implementation of IT projects, improved agility to adjust to ever changing market environment and reduce IT costs at the same time. Confidently make that path to finish the research according to a project plan and work schedule managing the risks and flaws.

Fig 2 :

 2 Fig 2: Cloud based ERP (Proposed system)

  Our proposed cloud based ERP flow chart has given bellows in the Fig 4.

  Fig 3: Traditional ERP flow chart[START_REF] Dr | llau@longwood.edu, developing a successful implementation plan for ERP[END_REF]

DISCUSSION AND ANALYSIS

Earlier ERP systems are implemented on the platform of premises of a company's server, thus hosted systems are at their own services. Proposed system allows implementing the ERP into a cloud that is managed by other vendor. Proposed system has an efficient option to choose business organization's own business flow as they recently acting on. More precisely we can say that, Cloud ERP has its own flow. But a business organization can also define their business by themselves. On the other hand ERP provider can define a standard business flow that can be followed by the business organization.

ERP packages are also has options to choose from cloud during ERP software selection by the business organization. Technically Cloud ERP is simple to deploy, organization need not to bear additional server and other dependent costs. It is also easy and quick to implement an ERP to a business organization. On the other hand, traditional ERP experience a challenge to deploying and maintenance.

Traditional ERP is very much controllable by the business organizations, because it is under their supervision. Again cloud ERP controlling depends on the support of the vendor. Vendor can ensure the control of ERP on be half of business organizations.

Traditional ERP has an easy accessibility both technically and user prospect. It does not depends on internet rather intranet. Alternatively, cloud ERP depends on internet. So if the internet bandwidth is low or it's technically departed then support cloud ERP is in trouble. Some specific and desirable quality of the traditional ERP and Cloud ERP are listed bellow in the Table 1.


Guillaume Marrelec

Alain Giron

Cumulants of multiinformation density in the case of a multivariate normal distribution

Keywords: dependence, information, cumulant-generating function, cumulants, mutual information, multiinformation

We consider a generalization of information density to a partitioning into N ≥ 2 subvectors. We calculate its cumulant-generating function and its cumulants, showing that these quantities are only a function of all the regression coefficients associated with the partitioning.

Introduction

Let X be a multivariate normal random variable with distribution f (x), and (X 1 , X 2 ) a partitioning of X into 2 subvectors with corresponding marginals f 1 (x 1 ) and f 2 (x 2 ). The information density relative to X and the partitioning (X 1 , X 2 ) is the random variable defined as (Polyanskiy and Wu, 2017, §17.1)

i d (X; X 1 , X 2 ) = ln f (X 1 , X 2 ) f 1 (X 1 ) f 2 (X 2 )
.

(1)

One of the key features of information density is that its expectation yields mutual information (Kullback, 1968, Chap. 1, §2;Polyanskiy and Wu, 2017, §17.1). In the present paper, we consider a partitioning of X into N ≥ 2 subvectors (X 1 , . . . , X N ) with corresponding marginals f 1 , . . . , f N and define multiinformation density as i d (X; X 1 , . . . ,

X N ) = ln f (X) N n=1 f n (X n )
.

1

The mean of this quantity

I(X; X 1 , . . . , X N ) = f (x) ln f (x) N n=1 f n (x n )
dx is itself a generalization of mutual information known under different names: total correlation [START_REF] Watanabe | Information theoretical analysis of multivariate correlation[END_REF], multivariate constraint [START_REF] Garner | Uncertainty and Structure as Psychological Concepts[END_REF], δ [START_REF] Joe | Relative entropy measures of multivariate dependence[END_REF], or multiinformation [START_REF] Studený | The multiinformation function as a tool for measuring stochastic dependence[END_REF]. Our interest in i d is driven by its close connection with mutual independence. Indeed, when the X n 's are mutually independent, multiinformation is classically equal to 0, but we also have i d ≡ 0 and Var(i d ) = 0 (see Appendix A), yielding other statistical markers of independence. By contrast, dependence between the X n 's is a multivariate phenomenon that multiinformation, as a one-dimensional measure, can only partially quantify. We expect i d to give a more detailed characterization of dependence, e.g., through its moments or cumulants.

We here focus on multivariate normal distributions. The family of multivariate normal distribution with given mean µ can be parameterized by either a covariance matrix or a concentration/precision (i.e., inverse covariance) matrix. Either parameterization shows multivariate distributions according to a certain perspective and emphasizes different features (e.g., Markov properties for the concentration matrix). In this context, we wished to investigate the existence of a natural way of parameterizing dependencies, i.e., a parameter that would emphasize the dependence properties of the distribution.

The core of the present paper is the following theorem:

Theorem 1 Let X be a d-dimensional variable following a multivariate normal distribution with mean µ and covariance matrix Σ. Partition X into N subvectors (X 1 , . . . , X N ), and set i d the corresponding multiinformation density. Then the cumulant-generating function of i d is given by

ln E e ti d = t I(X 1 ; . . . ; X N ) - 1 2 ln |I d -tΓ|, (2)
where

Γ = Σ diag(Σ 11 , . . . , Σ N N ) -1 -I d
is the block matrix whose diagonal blocks are equal to 0 and where each off-diagonal block (m, n) is the matrix of regression coefficients of X m on

x n Γ m|n = Σ mn Σ -1 nn , m = n. ( 3
)
The cumulants of i d are given by

κ 1 (i d ) = I(X 1 ; . . . , X N ) and κ l (i d ) = (l -1)! 2 tr Γ l , l ≥ 2.
This theorem is proved in Section 2. In Section 3, we investigate some consequences of this result. Section 4 is devoted to the discussion.

2 Proof of theorem

Cumulant-generating function

We partition µ and Σ in accordance with the partitioning of X, so that µ n is the expectation of X n and Σ mn the matrix of covariances between X m and X n . Multiinformation between the X n 's yields

I(X 1 ; . . . ; X N ) = 1 2 ln N n=1 |Σ nn | |Σ| .
From there, we can express multiinformation density as

i d = I(X 1 ; . . . ; X N ) + j d , (4)
where j d is defined as

j d = 1 2 (X -µ) t Φ(X -µ). (5)
and Φ as

Φ = diag(Σ 11 , . . . , Σ N N ) -1 -Σ -1 .
Here, diag(Σ 11 , . . . , Σ N N ) stands for the block-diagonal matrix with diagonal blocks equal to the Σ nn 's. The moment-generating function of j d yields

E e tj d = (2π) -d 2 |Σ| -1 2 e -1 2 (x-µ) t (Σ -1 -tΦ)(x-µ) dx.
Since it can be shown that Σ -1 -tΦ is positive definite at least in a neighborhood of t = 0 (see Appendix B), the integrand is proportional to a multivariate normal distribution with mean µ and covariance matrix Σ -1 -tΦ.

Integration with respect to x therefore yields

E e tj d = |Σ -1 | 1 2 |Σ -1 -tΦ| 1 2 = |I d -tΓ| -1 2 and ln E e tj d = - 1 2 ln |I d -tΓ|, (6)
where I d is the d-by-d unit matrix and Γ = ΣΦ the block matrix whose diagonal blocks are equal to 0 and where each nondiagonal block (m, n) is the matrix of regression coefficients of X m on x n given by (Anderson, 2003, Definition 2.5.1)

Γ m|n = Σ mn Σ -1 nn , m = n. (7)

Cumulants

The cumulants of i d can be calculated in closed form from those of j d and Equation ( 4) by noting that the first cumulant, κ 1 (i d ), is shift-equivariant, while the others, κ i (i d ) for i ≥ 2, are shift invariant (Kendall, 1945, §3.13). This leads to

κ 1 (i d ) = I(X 1 ; . . . ; X N ) + κ 1 (j d ) κ l (i d ) = κ l (j d ), l ≥ 2. ( 8
)
Now, the cumulants of j d can be easily computed as follows. Using the fact that |A| = e tr[ln(A)] [START_REF] Higham | Functions of matrices[END_REF], which, for a positive definite matrix, can be expressed as ln |A| = tr[ln(A)], we have from Equation ( 6)

ln E e tj d = - 1 2 tr [ln(I d -tΓ)] .
For t sufficiently small, we can perform a Taylor expansion of the log function around I d (Abramowitz and Stegun, 1972, Eq. 4.1.24), leading to

ln E e tj d = 1 2 tr ∞ il=1 (tΓ) l l = ∞ l=1 t l 2l tr(Γ l ).
Identification with the decomposition of the same function in terms of cumulants (Kendall, 1945, §3.12)

ln E e tj d = ∞ l=1 κ l t l l! , yields for the cumulants of j d κ l (j d ) = (l -1)! 2 tr(Γ l ). (9)
The same result could have been reached by using the fact that j d is a quadratic function of a multidimensional normal variate x, as evidenced in Equation ( 5), together with the expression of the cumulants of such functions [START_REF] Magnus | The exact moments of a ratio of quadratic forms[END_REF], Lemma 2). The cumulants of i d therefore yield

κ 1 (i d ) = I(X 1 ; . . . , X N ),
as expected, since the first cumulant is also the mean (Kendall, 1945, §3.14), and, for l ≥ 2,

κ l (i d ) = (l -1)! 2 tr Γ l .
In particular, the variance, which is equal to the second cumulant (Kendall, 1945, §3.14) is given by

Var(i d ) = κ 2 (i d ) = 1≤m<n≤N tr Σ mn Σ -1 mm Σ mn Σ -1 nn . (10)
In the even more particular case where all subvectors are 1-dimensional, we have

Var(i d ) = 1≤m<n≤N ρ 2 mn ,
where ρ mn is the usual correlation coefficient between X m and X n .

Consequences

We here investigate some consequences of the previous results: the particular case of partitioning into two subvectors, the irrelevance of the variances, and a graphical interpretation.

Partitioning into two subvectors

In the particular case where N = 2, multiinformation boils down to mutual information. The various powers of Γ can easily be calculated, yielding

Γ l = 0 χ l Υ l 0 for l odd and Γ l = χ l 0 0 Υ l for l even,
where we set

     χ l = Γ 1|2 Γ 2|1 . . . l factors Υ l = Γ 2|1 Γ 1|2 . . .,
with the relationship that χ l = Γ 1|2 Υ l-1 and Υ l = Γ 2|1 χ l-1 . For l odd, the trace of Γ l is equal to 0; for l even, it is equal to tr(χ l ) + tr(Υ l ), which can alternatively be expressed as

2 tr(χ l ) = 2 tr (Γ 1|2 Γ 2|1 ) l 2 or 2 tr(Υ l ) = 2 tr (Γ 2|1 Γ 1|2 ) l 2 .
In particular, the variance of i d is equal to

Var(i d ) = κ 2 (i d ) = tr Σ 12 Σ -1 22 Σ 21 Σ -1 11 . (11)
This quantity, which is a particular case of Equation ( 10), was introduced by [START_REF] Jupp | A general correlation coefficient for directional data and related regression problems[END_REF] as an extension of the classical correlation coefficient in the case of multidimensional variates, with application to directed data (Mardia and Jupp, 2000, §11.2). It is the sum of the squared canonical correlation coefficients between X 1 and X 2 (Anderson, 2003, Chap. 12;[START_REF] Jupp | A general correlation coefficient for directional data and related regression problems[END_REF].

If we furthermore assume that X 1 is a 1-dimensional vector, the cumulantgenerating function yields

ln E e ti d = - t 2 ln(1 -R 2 ) -ln(1 -t 2 R 2 ),
where R is the multiple correlation coefficient [START_REF] Anderson | An Introduction to Multivariate Statistical Analysis[END_REF], § 2.5.2)

R 2 = Σ 12 Σ -1 22 Σ t 12 σ 2 1 .
The cumulants are equal to 0 for l odd and to κ l

(i d ) = (l -1)! (R 2 )
l 2 for l even. In particular, the variance reads Var(

i d ) = R 2 .
Finally, if both X 1 and X 2 are assumed to be 1-dimensional vectors with correlation coefficient ρ, the cumulant-generating function reads

ln E e ti d = - t 2 ln(1 -ρ 2 ) -ln(1 -t 2 ρ 2 ),
with cumulants equal to 0 for l odd and κ l (i d ) = (l -1)! |ρ| l for l even. In particular, the variance yields Var(i d ) = ρ 2 .

Irrelevance of variances

Mutual information and multiinformation are both quantities that do not depend on the variance coefficients Σ kk 's. This result can be generalized to the cumulant-generating function of i d , and hence, its distribution, moments and cumulants. Indeed, let Σ = ∆R∆ be the decomposition of the covariance matrix

Σ such that ∆ = (∆ kl ) is a diagonal matrix with ∆ kk = √ Σ kk and R = (R kl ) is the correlation matrix with R kl = Σ kl / √ Σ kk Σ ll .
Block multiplication of ∆R∆ shows that we have Σ mn = ∆ mm R mn ∆ nn for any block Σ mn of Σ. Using the fact that (AB) -1 = B -1 A -1 for any two invertible square matrices A and B, we have

Γ m|n = ∆ mm R mn R -1 nn ∆ -1 nn = ∆ mm Γm|n ∆ -1
nn , where Γm|n is the matrix of regression coefficients obtained by application of Equation ( 7) to the correlation matrix R instead of the covariance matrix Σ. This result shows that Γ = (Γ m|n ) can be factorized into

Γ = ∆ Γ∆ -1 ,
where we set Γ = ( Γm|n ). This takes us to

|I d -tΓ| = |∆(I d -t Γ)∆ -1 | = |I d -t Γ|.
As a conclusion, we have that the cumulant-generating function of a multivariate distribution with covariance matrix Σ is the same as the cumulantgenerating function of a multivariate distribution with covariance matrix R, where R is the correlation matrix associated with Σ. This is a translation of the fact that i d does not depend on the variance coefficients.

Graphical interpretation

While simplification of the expression for the cumulants through the explicit calculation of Γ l is challenging in the general case, one can resort to a graphical interpretation of this matrix. Note first that the block (n, m) of Γ l is given by (Γ l ) mn = q 1 ,...,q l-1

Γ mq 1 . . . Γ q l-1 n l terms and tr(Γ l ) by tr(Γ l ) = N n=1 tr (Γ l ) nn = N n=1 tr   q 1 ,...,q l-1 Γ nq l-1 . . . Γ q 1 n   .
Consider then the directed and fully connected graph with N nodes {1, . . . , N }, an arrow from any m to any n = m (no self-connections), and corresponding (potentially matrix) weight Γ n|m . In this graph, a directed loop is a directed path that begins and starts at the same node. It is a k-loop if the directed path is composed of exactly k arrows. For any node n and integer k, let L → k (n) be the set of all directed k-loops starting and ending at node n and L → k the set of all k-loops. For any directed path p = (q 1 → • • • → q k ), define τ (p) as the trace of the product of the weights along p

τ (p) = tr Γ q k |q k-1 . . . Γ q 2 |q 1 .
With these notations, tr(Γ l ) can be interpreted as the sum of the values taken by τ along all directed l-loops starting and ending at every node of the graph

tr Γ l = N n=1 p∈L → l (n) τ (p).
It can also be seen as the sum of the values taken by τ along all directed l-loops

tr Γ l = p∈L → l τ (p).
See Figure 1 for an illustration of this interpretation. Note that the fact that 1-loops do not exist is interpreted as tr(Γ) = 0. This graphical interpretation is in particlar compatible with a partitioning into two subvectors. In that case, the corresponding directed graph only has two nodes and two arrows, and L → l = ∅ for l odd, which is in agreement with the fact that all cumulants of odd order are equal to zero.

Discussion

In the present manuscript, we introduced multiinformation density, a random variable that generalizes information density and whose expectation defines multiinformation. We focused on the case of a multivariate normal distribution and derived a closed form for its cumulant-generating function as well as its cumulants. We showed that the cumulant-generating function does not depend on the values taken by the variance coefficients of the covariance matrix, and that the computations required have a simple graphical interpretation. We also considered the special case of a partitioning into two subvectors, showing the relationship between our results and existing quantities.

Interestingly, the results show that the cumulant-generating function of multiinformation density is a function of a specific quantity, namely the block matrix Γ composed of all matrices of regression coefficients corresponding to the partitioning of X into (X 1 , . . . , X N ) as defined in Equation ( 7). This entails that the probability distribution of multiinformation density as well as all its moments are fully defined by Γ. In particular, it can be shown that mutual information and multiinformation themselves, as expectations, are functions of Γ only, namely (see Appendix C)

I(X 1 ; . . . ; X N ) = - 1 2 ln |I d + Γ|.
Going back to our question of knowing whether the multivariate normal distribution family could be parameterized in a natural way with emphasis on its dependence properties, it can be argued that Γ is a good candidate to this aim.

Generalization The expression of the cumulant-generating function of multiinformation in the case of multivariate normal distributions is quite simple. This simplicity is mostly a consequence of the stability of the multivariate normal family to most operations performed here: (i) the product of the marginals is also a multivariate normal distribution (this is strongly related to the fact that independence and uncorrelatedness are equivalent for multivariate normal distributions); (ii) the ratio of the joint distribution to its marginals takes a simple form that is again closely related to the multivariate normal family; and (iii) the exponentiation of t times the log of this ratio still has the form of a multivariate normal distribution. It would be of interest to determine such a simplicity would still hold in more general settings such as more general distribution families and more general functions of the ratio f X (X)/ f X i (X i ).

Regarding the type of family considered, a first step could be to consider the family of multivariate t distributions [START_REF] Kotz | Multivariate t Distributions and their Applications[END_REF]. In this case, f X (x) and f Y (y) are both multivariate t distributions [START_REF] Kotz | Multivariate t Distributions and their Applications[END_REF], §1.10) but f X (x) f Y (y) itself is not a multivariate t distribution, and the cumulant-generating function of the log of the resulting ratio does not have a simple closed form. Another, further step would be to consider elliptically contoured distributions (Anderson, 2003, §2.7), but again no simplification seems to occur.

Another potential generalization would be obtained by replacing the log function of multiinformation density with a quantity of the form

g f X (X) f X i (X i ) ,
in the spirit of f -divergences [START_REF] Csiszár | Eine informationstheoretische Ungleichung und ihre Anwendung auf den Beweis der Ergodizitat von Markoffschen Ketten[END_REF][START_REF] Csiszár | Information-type measures of difference of probability distributions and indirect observation[END_REF][START_REF] Ali | A general class of coefficients of divergence of one distribution from another[END_REF][START_REF] Vajda | On the f -divergence and singularity of probability measures[END_REF]. However, by replacing g(t) = ln(t) with another, more general form, we lose the key property that the log is the inverse of the exponentiation taken to compute the cumulant-generating function, which also simplifies calculations.

Future work Investigating the level of dependence between variables is still a thorny issue. Such an issue is usually tackled by investigating the properties of mutual information or multiinformation. We believe that considering multiinformation density instead of multiinformation (i.e., the random variable instead of its expectation) couldt contribute to a better characterization of dependence. For instance, the variance of i d , given in Equation ( 10), could be considered, in addition to or instead of multiinformation, to quantify the presence or absence of dependence. An important point to advance in this direction would be to provide estimators for the quantities obtained here. In the case of two variables, an estimator of this quantity, expressed in Equation ( 11), is readily available [START_REF] Jupp | A general correlation coefficient for directional data and related regression problems[END_REF]. Its generalization to more than two variables could yield a new tool to investigate mutual independence.

Another point of interest is the investigation of the behavior of multiinformation density in the case of large dimension. For instance, the asymptotic normality of this measure could be of interest. A quick derivation (see Appendix D) shows that, in the simple case of a homogeneous correlation matrix and a partitioning into 1-dimensional vectors, i d is not asymptotically normal. We hope that tackling this issue in the more general case will help better understand dependences within systems composed of many variables.

A Mutual independence, i d and multiinformation

We here show the equivalence between the following 1. The X n 's are mutually independent; 2. i d ≡ 0; 3. I(X 1 ; . . . ; X N ) = 0; 4. Var(i d ) = 0.

The equivalence between (1) and ( 2) is straightforward by definition of i d . If i d ≡ 0, then all its moments of i d are equal to 0, including its expectation (multiinformation) and its variance, leading to I(X 1 ; . . . ; X N ) = 0 and Var(i d ) = 0.

Since multiinformation is a Kullback-Leibler divergence, I(X 1 ; . . . ; X N ) = 0, entails that we have

f (x) = N n=1 f n (x n )
for all x, i.e., the X n 's are mutually independent. Finally, if Var(i d ) = 0, then i d is a constant, that is,

f (X) = k N n=1 f n (x n ).
The fact that f and the f n 's are distributions, and therefore must norm to 1, entails that we must have k = 1, that is, i d ≡ 0.

B Positive-definiteness of Σ -1 -tΦ

We need to show that Σ -1 -tΦ is a symmetric positive definite matrix in a neighborhood of t = 0. This matrix can be expressed as

Σ -1 -tΦ = (1 + t)Σ -1 -t diag(Σ 11 , . . . , Σ N N ) -1 .
As a difference of two symmetric matrices, it is also symmetric. Furthermore, since the two matrices in the right-hand side of the equation are positive definite they are diagonalizable in the same basis, i.e., there exists a nonsingular matrix F such that (Anderson, 2003, Theorem A.2.2)

F t Σ -1 F =    λ 2 1 . . . λ 2 d    and F t diag(Σ 11 , . . . , Σ N N ) -1 F = I.
Since Σ -1 is positive definite, we furthermore have λ 2 i > 0. Σ -1 -tΦ is therefore diagonalizable as well, with eigenvalues given by (1 + t)λ 2 i -t = (λ 2 i -1)t + λ 2 i , which is positive in a neighborhood of t = 0. Σ -1 -tΦ is therefore positive definite in a neighborhood of t = 0.

C Alternative expression of multiinformation

D Checking asymptotic normality

Let the correlation matrix R d be a d-by-d homogeneous matrix with parameter ρ, i.e., a matrix with 1s on the diagonal and all off-diagonal elements equal to ρ. such a matrix has two eigenvalues: 1+(d-1)ρ with multiplicity 1 (associated with the vector composed only of 1s) and 1 -ρ with multiplicity d -1 (associated with the subspace of vectors with a zero mean). Such a matrix is positive definite for

- 1 d -1 ≤ ρ < 1.
The expectation of i d is given by In particular, we have Var(i d ) = ρ 2 d(d -1)/2. For large d, we have κ l (i d ) ∼

(l-1)! 2 ρ l d l for l ≥ 2 and, in particular, Var(i d ) ∼ ρ 2 d 2 /2. To investigate the asymptotic normality of i d , we classically consider u = [i d -E(i d )]/ Var(i d ). Using the fact that the cumulant of order l is homogeneous of degree l, we obtain κ l (u) ∼ 2 l/2-1 (l -1)! = cste. If u were asymptotically normal, κ l (u) for l ≥ 3 would tend to 0 as d → ∞, which is not the case. As a consequence, u is not asymptotically normal.

Figure 1 :

 1 Figure 1: Graphical interpretation of tr(Γ l ). We consider the case N = 4 and l = 3. The directed 3-loop lp = (1 → 2 → 3 → 1) is represented with dark arrows. The value of τ on this loop is equal to τ (l) = tr(Γ 1|3 Γ 3|2 Γ 2|1 ). tr(Γ 3 ) is obtained by summing τ (p) over all 3-loops.

  For a decomposition of a multidimensional normal variable into several subvectors, multiinformation reads I(X 1 ; . . . ; X N ) calculateI d + Γ = I d + ΣΦ = I d + Σ diag(Σ 11 , . . . , Σ N N ) -1 -Σ -1 = Σ diag(Σ 11 , . . . , Σ N N ) -1 , leading to |I d + Γ| = |Σ diag(Σ 11 , . . . , Σ N N )

  1) ln(1 -ρ) + ln[1 + (d -1)ρ]} To compute the higher cumulants of i d , let U d the d-by-d matrix with all elements equal to 1. Using the fact that Γ = ρ(U d -I d ) together with U l d = d l-1 U d for l ≥ 2, we obtainΓ l = ρ l (-1) l I d + (d -1) l -(-1) l d U d tr(Γ l ) = ρ l (-1) l d + (d -1) l -(-1) l κ l (i d ) = (l -1)! 2 ρ l (-1) l d + (d -1) l -(-1) l .


Jacopo Baima

Jacek Goniakowski

Claudine Noguera

Alexey Koltsov

Jean-Michel Mataigne

Surface thermodynamics of silicate compounds: the case of Zn 2 SiO 4 (001) surfaces and thin films

Silicate compounds are ubiquitous in nature and display a vast variety of structures and properties. Thin silicate films may also form under specific conditions at interfaces between metals and silica. In the present study, we focus on zinc silicate and present a thorough density functional theory-based study of polar and non-polar (001) surfaces of various stoichiometry of its tetragonal polymorph t-Zn 2 SiO 4 . At the non-polar surfaces, the main features are the existence of the chain reconstruction at the ZnO termination, and the presence of unsaturated surface silanols at the SiO 2 termination. Stabilization of polar surfaces is provided by the formation of O 2- 2 peroxo groups, reduction of the surface or subsurface Si atoms or formation of Zn 2+ 2 groups, depending upon the surface stoichiometry. While the non-polar stoichiometric and ZnO rich terminations are the most stable in a large part of the accessible phase diagram, the SiO 2 termination is less stable due to the absence of siloxane group formation. We show that, while bulk Zn 2 SiO 4 is stable with respect to decomposition into the ZnO and SiO 2 oxides, the same is not true for ultra-thin films due to the fundamental difference of silicate and silica surface energies. Preliminary results show that a similar conclusion could be drawn Fe 2 SiO 4 . This study opens towards a deeper understanding and possible improvement of zinc adhesion at silica surfaces, of crucial industrial importance.

Introduction

Oxides have long since been used in various applications, ranging from thermal or electrical insulating barriers, to electronics, optoelectronics, or catalysis. Technologies continuously attempt to fine-tune their properties by means of nanostructuration or doping. On the one hand, controlled mixing of two oxides has often been considered as a promising way to engineer their properties, such as toughness, band gaps, or surface reactivities. On the other hand, uncontrolled spontaneous formation of ternary or mixed oxides at interfaces between materials may have non-negligible consequences for the properties of the junction and can be used as a lever for an optimization of its performances.

The role of such interface oxides is particularly interesting in the case of contacts between large gap oxides (such as silica or alumina) and late transition metals (such as zinc), where tuning the interface adhesion is a key challenge for the optimization of, e.g., anti-corrosive zinc coatings. Indeed, if galvanic zinc protection of iron-based materials has long proved its efficiency, 1 the oxidation and surface segregation of light strengthening elements (Al, Si, and Mn) in modern advanced high strength steels (AHSS), [2][START_REF]Advanced high strength sheet steels[END_REF][START_REF] Hu | [END_REF][5][6][7] dramatically reduce zinc adhesion. 8,9 We have recently shown that the most stable stoichiometric silica terminations, which expose surface siloxane rings interact very weakly with a zinc deposit 10 . Breaking these groups enables the formation of Zn-O bonds, enhances the interfacial interaction, and produces an overall more stable zinc/silica interface. However, it requires crossing substantial energy barriers and is not expected to spontaneously occur. In this context surface pre-hydroxylation appears as a promising route towards breaking surface siloxanes and improving the interface adhesion. 11 Aside application of inox buffer layers, as proposed for the similar zinc/alumina interfaces, 12,13 an alternative solution may rely on the formation of a mixed interfacial oxide phase, such as zinc silicate Zn 2 SiO 4 . [14][15][16] Such interfacial phase would impact the interactions occurring at the interface and, depending on the thermodynamic conditions, could enhance the adhesion through various pathways. The lower density of silicon at the silicate surface could, e.g., prevent the formation of siloxane rings, while the existing network of Zn-O bonds could naturally propagate towards the zinc deposit, thus allowing a stronger adhesion. A theoretical assessment of the differences between silicate and silica surface characteristics under different thermodynamic conditions could thus help defining appropriate process parameters to promote the adhesion of zinc coating on Si-alloyed steel grades.

In this context, the goal of the present study is twofold. First, in absence of existing data, we report a comprehensive mapping of Zn 2 SiO 4 surface energetics in a wide range of thermodynamic conditions and discuss the structural and electronic characteristics and relative stability of polar/non-polar and ZnO rich/SiO 2 rich terminations. Furthermore, we identify and discuss the main differences between the silicate and silica surface characteristics and link them to the presence/absence of surface silanol/siloxane groups. Finally, we show that, while bulk Zn 2 SiO 4 is stable with respect to decomposition into its ZnO and SiO 2 components, the same is not true for thin films under a critical thickness due to the fundamental difference of silicate and silica surface energies.

The paper is organized as follows. After presenting the details and settings of the computational approach in Sec. 2, in Sec. 3 we discuss bulk Zn 2 SiO 4 proprieties in reference to those of silica and zinc oxide. In Sec. 4 we first describe the alternative (polar and non-polar) surface configurations and then summarize their thermodynamic stability as a function of chemical conditions. Section 5 is devoted to a comparative analysis of silicate and silica surface energetics and to a discussion of its consequences for silicate thin film stability, before the conclusion.

Computational methods and settings

All calculations were performed within the Density Functional Theory (DFT) implemented in VASP (Vienna ab initio simulation package). 17,18 The dispersion-corrected GGA (optB86-vdW) exchange-correlation functional, [19][20][21] known to improve the description of adhesion characteristics, is used in view of future interface studies. We verified that the inclusion of the van der Waals correction results in a surface energy of the stoichiometric termination of Zn 2 SiO 4 larger by 0.23 Jm -2 with respect to a pure GGA functional. Similar variations are obtained for ZnO (0.21 Jm -2 ) and SiO2 (0.26 Jm -2 ). The trends, as well as the analysis of section 5 and conclusions, are therefore unaffected by the choice of the DFT functional. The interaction of valence electrons with ionic cores is described within the projector augmented wave (PAW) method. 22,23 The Kohn-Sham orbitals are developed on a plane-wave basis set with a cutoff energy of 400 eV (or 500 eV for relaxation of lattice parameters) and the self-consistent iterative solution of the electronic Hamiltonian is pursued until energy differences become less than 10 -6 eV. Atomic charges are estimated with the partition scheme proposed by Bader, 24,25 and atomic configurations are plotted with VESTA. 26 Bulk calculations While several zinc silicate polymorphs have been reported to form in thin films and at Si/ZnO or SiO x /ZnO interfaces [14][15][16] , we have focused on the tetragonal t-Zn 2 SiO 4 phase, which forms at a pressure of about 25 kbar, 27 and is only slightly less stable than the rhombohedral willemite. It is the second most stable phase of a series of five zinc silicate polymorphs formed when applying pressure to the willemite mineral, and has been already stabilized in the presence of interfacial strain. 14,16 The atomic structure of the t-Zn 2 SiO 4 polymorph (also referred to as Zn 2 SiO 4 -II) 16,27,28 is represented in Figure 1. In bulk calculations the lattice parameters and atomic coordinates were relaxed until the stress tensor and all forces become smaller than 0.01 eV Å -3 and 0.01 eV Å -1 , respectively. We find that with a Γ-centered 6×6×4 Monkhorst-Pack grid the calculated structural characteristics (lattice parameters, bond lengths, and angles) are converged to within 0.01 Å and 0.1 • .

Surface calculations We consider several stoichiometric and non-stoichiometric (001) terminations of t-Zn 2 SiO 4 , the relative stability of which depends of the values of the oxygen, zinc and silicon chemical potentials µ O , µ Zn and µ Si , respectively:

σ = (G slab Zn 2 SiO 4 -G bulk Zn 2 SiO 4 -∆N Si µ Si -∆N Zn µ Zn -∆N O µ O )/2S (1)
where G is the Gibbs free energy of symmetric slab or Zn 2 SiO 4 bulk and ∆N i is the number of excess atoms of specie i with respect to the Zn 2 SiO 4 stoichiometry. Since surface terminations are in equilibrium with the underlying bulk silicate, the chemical potentials are related through:

g bulk Zn 2 SiO 4 = 2µ Zn + 4µ O + µ Si (2)
where g bulk Zn 2 SiO 4 = G bulk Zn 2 SiO 4 /N is the Gibbs free energy per formula unit, implicitly depending on temperature and pressure. As a consequence, one of the chemical potential in Eq. 1 can always be eliminated by fixing N such that ∆N i = 0, and the surface energy can be expressed as a function of the other two. In the following, we will use µ O (which is accessible in the experiment and controls the polarity of the surface) together with either µ Zn or µ Si . We refer µ O to the total energy of a free oxygen molecule ∆µ O = µ O -1 2 E O 2 and the zinc and silicon chemical potentials to the total energies per atom of the bulk materials ∆µ Si,Zn = µ Si,Zn -E bulk Si,Zn . With this choice, ∆µ i 0 refers to conditions rich in the specie i (condensation of free oxygen molecules or formation of bulk Si/Zn) and the range of interest for this study is therefore ∆µ i ≤ 0.

Following standard procedures in ab initio thermodynamics, we approximate the free energy difference in Eq. 1 with the difference between the respective total energies at 0 K, obtaining:

σ (µ O , µ Zn ) = (E slab Zn 2 SiO 4 -E bulk Zn 2 SiO 4 -∆N Zn (E bulk Zn + ∆µ Zn ) -∆N O ( 1 2 E gas O 2 + ∆µ O ))/2S (3)
and σ (µ Si , µ O ) can be computed analogously.

As to limit the propagation of structural distortions inside the slabs and to improve the convergence with respect to their thickness, we use asymmetric slabs with one bare termination (labelled B in the following) and one fully hydroxylated termination (A in the following). The latter is chosen as the non-polar SiO 2 rich termination, described in Sec. 4, with 3 dissociated water molecules per surface unit cell.

In this case, the sum of the two surface energies is obtained as:

σ B (µ Zn , µ O ) + σ A (µ Zn , µ O ) = (E a-slab Zn 2 SiO 4 -E bulk Zn 2 SiO 4 -∆N Zn (E bulk Zn + ∆µ Zn ) -∆N O ( 1 2 E gas O 2 + ∆µ O ) -3E H 2 O )/S (4)
with E a-slab Zn 2 SiO 4 the energy of the asymmetric slab and E H 2 O the total energy of a water molecule. Since the evaluation of σ B requires the knowledge of σ A , the total energy of a symmetric slab with both terminations equivalently hydroxylated with 3 water molecules each is also calculated:

σ A (µ Zn , µ O ) = (E slab Zn 2 SiO 4 -E bulk Zn 2 SiO 4 -∆N Zn (E bulk Zn + ∆µ Zn ) -∆N O ( 1 2 E gas O 2 + ∆µ O ) -6E H 2 O )/2S (5)
Periodic slab replicas are separated by at least 10 Å of vacuum and we use dipole corrections to attenuate the effect of periodic boundary conditions in the direction normal to the slab surfaces. 17,29 The surface structures are obtained by relaxation of bulk cuts. In the cases in which the same surface stoichiometry can be obtained with several different initial geometries, we have relaxed the geometry starting from all initial configurations and retained the lowest energy one. All atoms were relaxed until forces become smaller than 0.01 eV Å -1 . With such settings 11 atomic layers (either Si or Zn 2 O 4 ) are sufficient to converge the surface energies to within 0.02 Jm -2 .

Bulk t-Zn 2 SiO 4

Similarly to willemite, bulk t-Zn 2 SiO 4 is formed by corner-sharing ZnO 4 and SiO 4 tetrahedra, with SiO 4 surrounded by ZnO 4 and each oxygen ion coordinated to one Si and two Zn atoms. Their structural properties are compared in Table 1. We remind that the experimental structure of t-Zn 2 SiO 4 is characterized under pressure p = 70 kbar, and this has to be taken into account when comparing with zero pressure results. Willemite has a larger unit cell with more nonequivalent sites and bonds, but its bond lengths and angles are only little dependent on the local environment. Most importantly, the difference in Si-O and Zn-O bond lengths between the two structures is relatively small (≤1.5%), and smaller than the difference between DFT results obtained with LDA and optB86-vdW exchange-correlation functionals. The O-Zn-O and O-Si-O angles inside the tetrahedra also retain similar values in both structures, with variations within 3%. The only small structural difference originates from different angles between the tetrahedra (Zn-O-Zn and Zn-O-Si), which differ by up to 10% between the two polymorphs. As these are the softest angles in the structure, they tend to bend at surfaces and inter- faces even if the connections between tetrahedra are not broken.

Bulk zinc silicate t-Zn 2 SiO 4 is 0.2 eV per formula unit more stable than bulk ZnO and SiO 2 , compared to 0.28 eV/f.u. measured for willemite. 31 As a consequence, at equilibrium with bulk SiO 2 and Zn (e.g., at the zinc/silica interface), t-Zn 2 SiO 4 may form in more oxygen lean conditions, i.e. for ∆µ O > -3.29 eV, compared to ZnO, which requires ∆µ O > -3.19 eV.

Structural and electronic properties of bulk t-Zn 2 SiO 4 are compared to those of bulk SiO 2 and ZnO in Table 2 and Figure 2. From the Bader charges it appears that the ionicity of the Si-O bond in t-Zn 2 SiO 4 is similar to that of SiO 2 , while the Zn-O bond is more ionic than in ZnO. In the LDOS, both the top of the valence band and the bottom of the conduction band are composed of hybridized Zn-O states, while occupied/unoccupied Si states are found at lower/higher energies, consistently with stronger and more ionic Si-O bonds. The increased ionicity of the Zn-O bond in the silicate accounts for a larger band gap compared to bulk ZnO (2.76 eV wrt 0.77 eV, the numbers which have only a relative significance, due to the known band gap underestimation when using semilocal DFT exchange-correlation functionals).

The bare (001) Zn 2 SiO 4 surfaces

As in most oxides, several alternative surface terminations may exist when a bulk is cut along a given orientation. Among them, some are non-polar and others are polar 32,33 , and variations in their stoichiometry may further increase the number of configurations to consider. In the following we will successively present the characteristics of the non-polar and polar t-Zn 2 SiO 4 (001) sur-

Non-Polar surfaces

Along the (001) direction, the repetitive unit cell is composed of the .../Zn 2 O 4 /Si/... sequence of atomic layers, with a negligible height difference between the Zn and O atoms (see Fig. 1). These layers have a formal charge of -4 for Zn Moreover, at the SiO 2 termination, due to the low density of silicon atoms and the large distance between the surface silanols SiO, no siloxane rings can form contrary to pure silica surfaces 10 .

Due to the lower coordination of surface atoms, the electronic structure of the three terminations differs from the bulk one. The reduction of the surface Madelung potential induces a reduction of the band gap of 0.5-0.7 eV and a more covalent character of the surface bonds (smaller Bader charges in Tab. 3). At the SiO 2 termination, the surface states forming the band gap have a slightly enhanced Si-O character (Fig. 4), while in the other two cases the Zn-O character is dominant, similarly to the bulk silicate.

Polar surfaces

Six polar terminations can be obtained by addition or subtraction of an oxygen atom to the three non-polar surfaces: the SiO 3 , ZnO ever, at the Zn 2 O 3 termination, due to their strong hybridization with Zn states, the band gap is larger than at the corresponding Zn 2 O 2 non-polar termination. At oxygen lean terminations, two additional electrons per surface cell are needed to cancel the polarity. They are systematically localized at surface cations, but this takes different forms in the three cases. At the SiO termination, the surface silicon atom hosts the excess electrons. Silicon states are shifted towards the gap edges, so that the reduction of E g is modest. At the Zn termination, the electrons are mostly localized on the sub-surface silicon atoms, with a smaller contribution on the surface zinc atom. This results in a nearly vanishing band gap. Finally, the Zn 2 O termination undergoes a significant reconstruction with the formation of a surface Zn 2+ 2 specie. The top of valence bands only involves zinc states, which also results in a large reduction of the band gap.

Surface stability

As stated in Sec. 2, the relative stability of the nine surfaces depends on the values of the excess chemical potentials ∆µ i . The top and bottom panels of Fig. 5 show the most stable surface configurations as a function of (∆µ O , ∆µ Zn ) and (∆µ O , ∆µ Si ), respectively. They represent two alternative projections of the two dimensional plane defined by the bulk silicate stability condition, Eq. 2. The solid diagonal lines represents the conditions ∆µ Si = 0 (upper panel) or ∆µ Zn = 0 (lower panel). Thus, the upper triangle corresponds in both cases to the area where all ∆µ i < 0.

Under most physical conditions, the non-polar surfaces are the most stable, which is consistent with the large energy cost of accommodating the excess charges at the polar terminations. Among them, the Zn 2 O 2 one is the most stable in zinc rich conditions, the SiO 2 one in silicon rich conditions, and the stoichiometric ZnO one at intermediate values of the chemical potentials. The existence or absence of surface reconstructions plays an important role. Indeed, the stoichiometric ZnO termination features a significant reconstruction, but the strain on bond lengths and angles prevents it from being stable in a wider range of conditions. Similarly, the absence of siloxane rings at the SiO 2 -terminated surface results in its lower stability and prevents it from being more competitive with the ZnO rich terminations.

Polar terminations are only stable under relatively oxygen rich or oxygen lean conditions, Fig. 5. However, contrary to the case of silica where the SiO and SiO 3 terminations are only marginally stable, 10 these terminations can more easily form (for ∆µ O -0.75 eV or ∆µ O < -2.9 eV) at silicate surfaces, producing potentially more reactive surface configurations. At variance, the Zn and ZnO 2 terminations, obtained by addition/removal of one oxygen to/from the stoichimetric ZnO surface, are not thermodynamically stable under any of considered conditions. As expected, zinc rich conditions favor the Zn 2 O termination, oxygen and silicon rich conditions favor the SiO 3 termination, and oxygen and zinc rich environments favor the Zn 2 O 3 one.

Discussion

In the following, we first highlight the main differences between the surface energetics of Zn 2 SiO 4 and pure SiO 2 and link them to the different nature of the exposed surface species. Then, we discuss the stability of Zn 2 SiO 4 in the limit of ultra-thin film.

While Figure 5 gives a comprehensive mapping of most stable silicate surfaces, a complementary picture can be obtained from plots in which one additional condition is imposed to the chemical potentials. Possible choices may involve either an equilibrium with bulk zinc (∆µ Zn = 0) or bulk silica (∆µ

Si + 2∆µ O = E bulk SiO 2 -E bulk Si -E O 2 )
, relevant for zinc silicate formed at the interface between zinc and silica. The energies of all surface terminations can then be plotted as a function of a single chemical potential ∆µ O , as shown in Fig. 6.

When a silicate surface is in equilibrium with bulk zinc (top panel of Fig. 6), the most stable termination is always one of the three ZnO rich ones: Zn 2 O, Zn 2 O 2 , or Zn 2 O 3 . Similarly, when at equilibrium with bulk silica (bottom panel), the most stable termination in a wide range of chemical potential µ O is the stoichiometric ZnO one, the ZnO rich Zn 2 O 2 termination being very close in energy. Under extreme values of the oxygen chemical potential, the stable polar terminations are also ZnO rich. We note that the calculated energies of the stoichiometric ZnO and Zn 2 O 2 terminations (1.3 Jm -2 , when in equilibrium with bulk ZnO) are hardly larger than that of the most stable stoichiometric wurtzite ZnO (10 10) surface (1.1 Jm -2 ). Conversely, regardless of the precise thermodynamic conditions, the three SiO 2 rich terminations (SiO, SiO 2 , and SiO 3 ) are significantly less stable. Interestingly, in equilibrium with bulk silica, the energy of the stoichiometric silicate SiO 2 termination (1.8 Jm -2 ) is substantially larger than that of the stoichiometric β -cristobalite (001) SiO 2 termination (1.0 Jm -2 ). Indeed, in pure silica, the corresponding stability diagram, Fig. 7, 10 shows that the SiO 2 surface energy is greatly reduced upon formation of surface siloxane rings (D 2 -SiO 2 termination). However, since such rings cannot form at silicate surfaces because of the lower density of surface Si-O silanol groups, the energy of the silicate SiO 2 termination is rather comparable to that of the much less stable unreconstructed silica surface (SiO 2 termination in Fig. 7) and remains higher than these of the ZnO and Zn 2 O 2 terminations.

Beyond the lower stability of SiO 2 rich surfaces, we will show that the absence of silanol polymerization may also produce a thermodynamic bias for the decomposition of silicate nanoobjects (e.g., thin films) into silica and zinc oxide. In the bulk, the mixing energy E bulk mix is estimated from the total energies of bulk Zn 2 SiO 4 , SiO 2 , and ZnO materials as:

E bulk mix = E bulk Zn 2 SiO 4 -E bulk SiO 2 -2E bulk ZnO (6)
It is negative, E bulk mix = -0.2 eV/f.u., when t-Zn 2 SiO 4 is referred to β -cristobalite SiO 2 and wurtzite ZnO, showing that t-Zn 2 SiO 4 is stable with respect to the two constituting oxides. For the most stable silicate structure (willemite), E bulk mix is even more negative and closer to the experimental of -0.28 eV/f.u.. In analogy to its evaluation in oxide clusters 34 , the mixing energy (per formula unit) in thin films E f ilm mix can be derived from the total energy of a silicate slab, referred to SiO 2 and ZnO slabs containing as a whole the same number of ions and exposing the same number of surfaces. This can be obtained by calculating slabs containing n, 2n, and 4n formula units for Zn 2 SiO 4 , SiO 2 , and ZnO respectively, and averaging the binary oxide energies:

E f ilm,n mix = [E f ilm,n Zn 2 SiO 4 - 1 2 (E f ilm,2n SiO 2 + E f ilm,4n ZnO )]/n (7)
In the explicit DFT calculations of the film mixing energy we have systematically used slabs which expose the smallest surface cell area of the most stable surface, as to minimize their total energies for the given number of formula units. For n = 4 (silicate film thickness of about 7 Å) we find a positive E f ilm mix = 1.0 eV/f.u., thus unambiguously showing that at this thickness the silicate film is unstable with respect to decomposition into (10 10)-oriented ZnO and β -cristobalite SiO 2 (001) films. This result can be rationalized by considering the different surface characteristics of silica and silicate discussed above. Indeed, if the total energy of a film E f ilm,n is approximated as E f ilm,n ∼ n E bulk + 2 S E sur f , with S the unit cell surface area and E sur f the energy of a semi-infinite surface, the film mixing energy, Eq. 7, reads: = 0.9 eV/f.u., consistent with the above explicit computational estimation.

E f ilm,n mix ∼ E bulk mix + δ E sur f /n (
According to the equation 8, the silicate film is thus unstable with respect to decomposition into the constituting oxide films for thicknesses below n = -δ E sur f /E bulk mix ∼ 20 f.u.. The positive value of δ E sur f , which drives such silicate instability in the limit of small film thickness, reflects the difference between the surface energies of the three materials. To a large extent, it is due to the substantial difference between pure SiO 2 and Zn 2 SiO 4 surface energies, driven by the absence of siloxane groups at silicate surfaces.

As to put a broader perspective to these findings, we have attempted a preliminary estimation of analogous decomposition of iron silicate films. To this goal we have considered iron silicate Fe 2 SiO 4 in its spinel phase, 35,36 which is structurally similar to the t-Zn 2 SiO 4 one and is favored over fayalite under higher pressures. Considering the decomposition of a Fe 2 SiO 4 film into pure SiO 2 (001), Fe 2 O 3 (0001), and Fe(110) films, the mixing energy involves the following surface energy term:

δ E sur f = 2S Fe 2 SiO 4 E sur f Fe 2 SiO 4 -2 3 (S SiO 2 E sur f SiO 2 + S Fe 2 O 3 E sur f Fe 2 O 3 + S Fe E sur f Fe ) (10)
With the calculated surface energies equal to E sur f Fe =2.12 Jm -2 , and surface areas of S Fe 2 SiO 4 = 33.8 Å 2 , S Fe 2 O 3 = 25.4 Å 2 , and S Fe = 5.9 Å 2 , we obtain δ E sur f = 1.0 eV. Although smaller than in the case of Zn 2 SiO 4 , the positive value of δ E sur f clearly shows that also in this case the exposed surfaces reduce the stability of a thin silicate film with respect to its constituents. As a reference, the experimental mixing energy of bulk fayalite is equal to -0.2 eV/f.u. only. Interestingly, the decomposition of silicate film may not occur in the case of Mn 2 SiO 4 , for which the bulk term is much more negative (-0.51 eV/.f.u. with respect to MnO and SiO 2 ) 31 and the positive surface term is even smaller due to the relatively low MnO(100) surface energy (0.7 Jm -2 within DFT-GGA) 37 .

Conclusions

We have performed a thorough study of the t-Zn 2 SiO 4 bulk and (001) surfaces, using a DFT approach. We have considered all possible surface compositions in a (1 × 1) unit cell, leading to alternative non-polar and polar terminations. We have analyzed surface electronic and energetic characteristics and have highlighted the existence or absence of surface reconstructions.

At the non-polar surfaces, the low coordination of surface atoms induces little modifications of the electronic structure compared to the bulk silicate. The main features are the existence of the chain reconstruction at the ZnO termination, and the absence of siloxane groups at the SiO 2 termination. At the polar surfaces, polarity is healed thanks to the formation of O 2- 2 peroxo groups under oxygen excess, while under oxygen deficiency, the compensating charges are provided either by reduction of the surface or subsurface Si atoms or by the formation of Zn 2+ 2 groups. These processes mainly impact the nature of the band edges and the gap width.

The relative stability of the various surface terminations has been determined as a function of the oxygen, zinc, and/or silicon chemical potentials. We find that the non-polar stoichiometric and ZnO rich terminations are the most stable in a large part of the accessible phase diagram. We assign the relatively low stability of the non-polar SiO 2 termination to the absence of reconstruction of its surface silanols into siloxane rings.

As a first step towards the understanding of zinc silicate formation at the interface between zinc and silica, we have determined the mixing properties of ZnO and SiO 2 to form Zn 2 SiO 4 both in the bulk and in thin silicate films. While bulk Zn 2 SiO 4 appears stable with respect to decomposition into the two constituting oxides, we find that the same is not true for thin films under a critical thickness. We assign the thin film instability to the fundamental difference between the surface energies in silica and in silicates due to the presence/absence of silanol reconstruction into siloxane rings. Preliminary calculations show that a similar conclusion could be drawn also for the Fe 2 SiO 4 silicate.

This study opens the way towards the analysis of formation of interface zinc silicate and its consequences for zinc adhesion at silica surfaces under various thermodynamic conditions, of crucial industrial importance.

a

  CNRS, UMR 7588, Institut des Nanosciences de Paris, F-75005 Paris, France. E-mail: jacek.goniakowski@insp.jussieu.fr b Sorbonne Université, Institut des Nanosciences de Paris, UMR 7588, INSP, F-75005 Paris, France. c ArcelorMittal Maizières Research, voie Romaine, F-57280, Maizières lès Metz, France.

Fig. 1

 1 Fig. 1 Bulk structure of t-Zn 2 SiO 4 . Si, Zn, O atoms are represented by large blue, large grey, and medium red balls, respectively.

Fig. 2

 2 Fig. 2 Local density of states (LDOS) of bulk SiO 2 (top), ZnO (center), and t-Zn 2 SiO 4 (bottom) projected on zinc (black), silicon (blue) and oxygen (red) atoms.

2 O 4

 24 and +4 for the Si layer, showing that the (001) orientation is polar. To cancel this polarity, the formal charge of the surface layers has to be reduced to ±2. This is obtained when the number of oxygen atoms in the topmost layer exceeds by one the number of zinc atoms. The three terminations for which this condition is fulfilled are Zn 2 O 3 /Si/Zn 2 O 4 /Si/..., ZnO 2 /Si/Zn 2 O 4 /Si/... and O/Si/Zn 2 O 4 /Si/... In order to label these surfaces in a way that defines their composition as well as their polarity (charge neutrality), we formally assign one oxygen atom from the surface layer to the sub-surface Si. With this choice, the charge neutral (non-polar) terminations given above are labeled Zn 2 O 2 , ZnO, and SiO 2 (see Fig.3). Of these three, the ZnO termination is stoichiometric with respect to a dipole-free bulk unit cell, and can be used to build a symmetric stoichiometric ZnO-terminated slab. The Zn 2 O 2 and SiO 2 terminations are ZnO rich and SiO 2 rich, respectively. At the ZnO termination, only weak Zn-O bonds are cut upon surface formation. Moreover, this termination undergoes a significant reconstruction which allows two new Zn-O bonds to be formed. The surface atoms arrange themselves in Zn-O-Si-O chains with a concomitant distortion of bond angles. At variance, if the Zn 2 O 2 and SiO 2 terminations have a small total number of broken bonds, strong Si-O bonds are cut upon surface formation.

Fig. 3

 3 Fig. 3 Atomic structures of the alternative terminations of the (001) surface. Left to right: oxygen lean to oxygen rich. Top to bottom: SiO 2 rich to ZnO rich. In each row, the central configuration is non-polar. Si, Zn, and O atoms are represented by large blue, large grey, and medium red balls, respectively.

2

 2 and Zn 2 O 3 oxygen rich terminations, and the SiO, Zn and Zn 2 O oxygen lean terminations, which are shown in Fig. 3. Similarly to the non-polar case, SiO x (x = 1, ..., 3) terminations are SiO 2 rich and Zn 2 O x (x = 1, ..., 3) terminations are ZnO rich. At all oxygen rich terminations, polarity compensation requires the presence of two holes per surface unit cell. The holes are accommodated via the formation of surface peroxo groups, of O 2- 2 formal charge. The two oxygen atoms share an equivalent local environment at the SiO 3 and Zn 2 O 3 terminations, with either Si-O or Zn-O bonds.In the ZnO 2 case, one oxygen atom is bonded to a surface silicon and the other to two zinc atoms, which results in an asymmetry between their Bader charges. In most cases, nearly pure peroxo states compose the band gap edges (seeFig 4), which reduces its width with respect to the bulk one. How-

Fig. 4

 4 Fig. 4 Local density of states (LDOS) at the (001) surface of t-Zn 2 SiO 4 projected on surface zinc (black), silicon (blue), and oxygen atoms (orange for peroxo groups, red for other surface oxygen atoms) at the nine terminations. Left to right: oxygen lean to oxygen rich. Top to bottom: SiO 2 rich to ZnO rich.

3 Fig. 5

 35 Fig. 5 Surface stability of t-Zn 2 SiO 4 : most stable surface termination as a function of oxygen and zinc (top) and oxygen and silicon (bottom) chemical potentials. Solid diagonal lines represent ∆µ Si =0 (top) and ∆µ Zn = 0 (bottom). The dashed (dotted) line in the top (bottom) panel corresponds to the equilibrium with bulk ZnO (SiO 2 ).

Fig. 6

 6 Fig. 6 Calculated surface energies of the different terminations of the t-Zn 2 SiO 4 (001) surface as a function of oxygen chemical potential ∆µ O , obtained for equilibrium with either bulk zinc (top) or with bulk SiO 2 (bottom). Solid vertical lines represent ∆µ Si = 0 (top) and ∆µ Zn = 0 (bottom) and the dashed line represents equilibrium with bulk ZnO (see text).

Fig. 7

 7 Fig. 7 Calculated surface energies of the different terminations of the SiO 2 (001) surface as a function of oxygen chemical potential ∆µ O . The D 2 -SiO 2 termination is the reconstructed non-polar termination with two-membered siloxane rings. Adapted from Ref. 10.

Fe 2

 2 SiO 4 = 1.21 Jm -2 , E sur f Fe 2 O 3 = 1.52 Jm -2 , and E sur f

Table 1

 1 Comparison of the computed structural properties [bond lengths (Å), angles ( • )] of t-Zn 2 SiO 4 with x-ray scattering data obtained under pressure p = 70 kbar and with the willemite structure.

		Si-O	Zn-O	Zn-O-Zn	Zn-O-Si	O-Zn-O	O-Si-O
	t-Zn 2 SiO 4 (optB86-vdW)	1.64	1.98	116.7	118.0/123.5	101.7/104.4/117.2	108.0/112.4
	t-Zn 2 SiO 4 (LDA) 16	1.52	1.93	109.0	118.6/125.0	96/111.2/114.5/120.4	101.5/126.8
	t-Zn 2 SiO 4 (exp) 28	1.62	1.98	115.2	117.5/124.8	100.2/106.9/115.5/117.3 107.7/113.2
	Willemite (LDA) 16	1.63/1.64	1.92-1.95	105.6/111.8/112.2	117.1/126.7/128.6/131	105.6-112.2	107.2-111.2
	Willemite (exp) 30	1.63/1.64 1.95/1.98 106.3/109.7/112.3 117.5/120.9/126.0/132.6	103.2-117.0	106.1-110.8

Table 2

 2 Calculated Bader charges Q (e), band gaps E g (eV), and anion-cation bond lengths d (Å) in bulk t-Zn 2 SiO 4 , β -cristobalite SiO 2 , and wurzite ZnO.

		t-Zn 2 SiO 4	SiO 2	ZnO
	Q O (e)	-1.43	-1.59 -1.16
	Q Zn (e)	1.29	-	1.16
	Q Si (e)	3.12	3.18	-
	E g (eV)	2.76	5.85	0.77
	d Si-O (Å)	1.64	1.63	-
	d Zn-O (Å)	1.98	-	1.99
	faces.

Table 3

 3 Calculated Bader charges Q (e) of surface atoms and band gaps E g (eV) for each surface termination, compared with the corresponding values in bulk zinc silicate.

		SiO	SiO 2	SiO 2	Zn	ZnO	ZnO 2	Zn 2 O	Zn 2 O 2	Zn 2 O 3	Bulk
	Q O,peroxo (e)	-		-0.78			-0.58/-0.85			-0.62
	Q O,other (e)	-1.40 -1.35	-1.42	-1.31/-1.38 -1.43/-1.36	-1.42	-1.40	-1.15/-1.41	-1.39	-1.43
	Q Zn (e)	1.24	1.26	1.27	1.02	1.30	1.28	0.51/0.85	1.23	1.24	1.29
	Q Si (e)	1.65	2.87	3.11	1.97	3.08	3.08	3.09	3.10	3.09	3.12
	E g (eV)	2.18	2.04	1.88	0.1	2.09	1.85	1.75	2.16	2.38	2.76

  Zn 2 SiO 4 -S SiO 2 E Zn 2 SiO 4 =1.34 Jm -2 , E SiO 2 =0.97 Jm -2 , and E ZnO =1.12 Jm -2 , and the corresponding surface areas of S Zn 2 SiO 4 = 50 Å 2 , S SiO 2 = 48 Å 2 , and S ZnO = 17 Å 2 , we obtain a positive surface term δ E sur f = 4.2 eV and E

		sur f SiO 2 -S ZnO E	sur f ZnO	(9)
	With the calculated surface energies equal to E	sur f
	sur f	sur f
	f ilm,n=4
	mix

8)

It involves the bulk mixing energy E bulk mix and a surface term δ E sur f which reads:

δ E sur f = 2S Zn 2 SiO 4 E sur f

| 1-9

Acknowledgments

J. Baima acknowledges a post-doctoral grant from ArcelorMittal Maizières Research. This work was supported by HPC resources from GENCI-IDRIS (Grant 2017-A0010906855).