Sabrine Kardous

Timothy F L Mckenna

Nida Sheibat-Othman

Thermodynamic effects on grade transition of polyethylene polymerization in fluidized bed reactors

Keywords: Grade transition, fluidized bed reactor, condensed mode cooling, thermodynamics, polyethylene

An off-line dynamic optimization procedure is employed to optimize the transition between different grades of linear low density polyethylene in a fluidized-bed reactor. This type of reactor is frequently operated under condensed mode, which consists of injecting induced 2 condensing agents (ICA) to absorb part of the reaction heat. However, the presence of ICA affects the solubility of monomers in the polymer, so it is important to account for this effect in a grade transition optimization strategy. A kinetic model is combined with a thermodynamic model based on the Sanchez-Lacombe equation of state to describe the grade transitions.

Simplified correlations are then suggested to predict the impact of ICA on ethylene and comonomer solubility in a quaternary system. The results highlight the importance of the thermodynamic model during grade transition.

Introduction

Polyethylene (PE) is the most widely produced polymer in the world. Among the different types of PE, linear low density polyethylene (LLDPE) occupies an important position with around 30% of the global PE production in 2018. [START_REF] Hutley | Polyolefins-the history and economic impact[END_REF] Most processes used to make LLDPE are gas-phase processes which provide several advantages over slurry processes. Indeed, difficulties related to mass transfer limitations and the dissolution of the amorphous polymer in a diluent, as well as fouling in slurry processes are avoided in gas-phase processes. Gas-phase processes are adequate for multipurpose production and permit the production of a wide range of PE grades.

Among gas-phase processes, fluidized-bed reactors (FBRs) are far and away the most widely used reactors for the production of LLDPE because they are the only reactors that can remove enough heat in the gas phase and thus allow the production of large amounts of polymer. [START_REF] Soares | Polyolefin Reaction Engineering[END_REF] In order to further enhance heat transfer and increase productivity, condensed mode cooling is frequently employed, where induced condensed agents (ICAs), which are typically alkanes such propane or isomers of butane, pentane or hexane, are injected in either liquid or vapor form. [START_REF] Mckenna | [END_REF]4] The heat of vaporization and/or increase in the heat capacity of the vapor phase in the reactor absorb a significant amount of the reaction heat and improve the control of the reactor temperature. However, it has also been observed that when the polymer particles are swollen by an alkane or an alkene, the reaction rate can change significantly due to the so-called cosolubility effect. Indeed, the presence of a hydrocarbon heavier than ethylene enhances the solubility of the latter in the amorphous phase of the polymer, thereby contributing to a higher rate of polymerization, while the lighter hydrocarbons play the role of anti-solvent for the heavier ones. [5,6] Therefore, the presence of ICA increases the ethylene concentration in the particles, leading to a higher reaction rate, while ethylene is expected to act as an anti-solvent for the ICAs (and eventually for comonomers like 1-butene or 1-hexene). This is expected to impact the properties of the final product such as its molecular weight and density. Therefore, it is important to account for these effects in the process model, especially in model-based optimization or control strategies.

It is quite common to produce several grades in the same polymerization plant in order to obtain a PE with different density, molecular weight and polydispersity index required in the various applications of PE. [7] Frequent transitions between these grades are usually needed to suit the market demand and reduce the storage cost. Due to the long residence time in FBRs, compared to tubular or loop reactors for instance, the flow rates should be optimized wisely to ensure attaining the new set-point in a short time, thus reducing the amount of transition product. When condensed mode is employed, the transitions might be more complex since the sorption/desorption dynamics of the different species can change the behavior of the system.

The employment of an adapted control of the transition is thus essential in order to optimize the economic yield while ensuring the security of the operations. 4 Debling et al. [7] studied the effect of different parameters on grade transition of solution, slurry, bulk and gas-phase polyolefin reactions in commonly used reactors, including horizontal or vertical stirred beds, loop and FBRs. They indicated the residence time distribution of the different components to be a determining factor in the speed of grade transition and summarized the procedures employed to speed the transition in FBRs, such as de-inventorying the reactor content or venting/overshooting the gases at the beginning of the transition. However, they did not investigate the effect of ICA on the residence time of the reactor. Rahimpour et al. [8] also indicated that partial venting of the reactor, composing a new gas phase and reducing the bed level reduce the quantity of transition product in PE FBRs. They highlighted that such so-called semi-continuous strategy was necessary in some situations in order to keep the reactor temperature between the gas dew point and the polymer melting point (to avoid agglomeration of the particles), which could not be achieved with the continuous strategy (i.e. by controlling only the flow rates). Note that the flow rates employed during the transition were those used for the final grade, which were calculated by solving the model equations under steady state conditions, and identifying the boundary conditions to be implemented for each new grade. However, numerous works indicated that the flow rates of the final grade do not necessarily ensure the best transition, and suggested the employment of dynamic optimization or control algorithms to ensure better transitions. [START_REF] Ohshima | Grade Transition Control for an Impact Copolymerization Reactor[END_REF]10] For the particular problem of grade transition in FBRs, most works are based on offline optimization due to the long calculation time and the complexity of problem formulation.

McAuley et al. [11,12] were the first to investigate dynamic optimization of grade transition of PE in gas-phase FBR. They provided a kinetic model for copolymerization, correlations for the final properties based on patent data (i.e. melt index and polymer density), and modelled the FBR as a continuous stirred tank reactor (CSTR) due to its high recycle ratio and low single pass conversion. The suggested control variables were the flow rates of hydrogen and comonomer (1-butene or 1-hexene), as they directly affect the polymer molecular weight and density. Afterwards, optimization strategies were proposed for different types of processes, for instance for slurry high density polyethylene (HDPE) processes composed of two loop reactors [13] or two CSTRs. [14] Furthermore, different improvements in the optimization approaches were suggested. For instance, Chatzidoukas et al. [15] proposed a mixed integer dynamic optimization approach to realize a closed-loop control in a fluidized-bed reactor. Nystrom et al. [16] employed a comparable approach based on dynamic optimization combined to a mixed-integer linear problem related to the sequencing, and solved these decoupled problems by iteration. Bonvin et al. [17] proposed to employ a measurement-based approach by tracking the necessary conditions of optimality, for instance based on run-to-run basis, in order to correct for modelling mismatch in a homopolymerization process.

Regarding closed-loop control, it was usually considered using algorithms based on an optimization criterion, such as model predictive control (MPC). The closed-loop character of MPC makes it more robust to modelling errors than open-loop dynamic optimization. But, in order to allow its online implementation in FBRs, part of the optimization is usually solved offline. For instance, Wang et al. [18] combined an offline optimizer and a nonlinear MPC, where the optimal feed rates were calculated offline and the MPC allowed minimizing the modeling error and updating the feed rates. A shrinking horizon nonlinear model predictive control with expanding horizon least-squares estimation was also implemented to control the grade transition in FBRs. [START_REF] Wang | Optimal Grade Transitions in a Gas-phase Polymerization Fluidized Bed Reactor[END_REF] Among all the cited works, in terms of methodology, dynamic optimization-based policies were the most widely used for gas-phase processes, and they will therefore be employed in this work. Indeed, the optimization criterion is more flexible and can be tuned to optimize instantaneous or cumulative properties during transition, or the transition time. In addition, the previous 6 literature analysis highlights that the parameters that most affect the grade transition are the residence time (distribution) and the hydrogen and comonomer flow rates. However, the thermodynamic interactions that are due to the use of a condensing agent and/or comonomer were not considered. This work is focused particularly on condensed mode, and will explore the implications of using a more representative thermodynamic model than most works that takes into account the interactions between the different species. It is clear that simply using additive solubilities will lead to erroneous conclusions about reaction rates and product properties. It is thus important to understand whether or not this is an important part in optimizing grade transitions.

In this work, the grade transition of PE copolymers in a gas-phase FBR operating under condensed mode is considered. First the effect of ICA (n-hexane or iso-butane) on the absorption of monomer (ethylene) and comonomer (1-butene or 1-hexene) is investigated using a model based on the Sanchez-Lacombe equation of state (SL EOS) and experimental data from literature. [20,21] Since no sufficient experimental data are available in the open literature for quaternary systems (i.e. a copolymerization in presence of an ICA), simplifying correlations are proposed in order to allow for fast prediction of the co-solubility effects in a quaternary system. [21] The thermodynamic correlations are then used to calculate equilibrium solubilities for two copolymerization systems of ethylene with -olefins. The thermodynamic predictions are combined with kinetic copolymerization models to model the dynamic behavior of the FBR, which is assumed to behave like a single-phase CSTR. The model is valid in the super-dry upper compartment of the FBR, containing only gas and polymer. This dynamic model is finally used within a dynamic optimization strategy to optimize the transitions between different grades of LLDPE.

Gas-phase catalytic ethylene co-polymerization model

Thermodynamic model

The SL EOS has been used frequently to predict the thermodynamic behavior in binary and ternary systems (polymer plus one or two penetrants respectively). [22,23] However, as discussed by McKenna [START_REF] Mckenna | [END_REF] , solubility data for systems of two penetrants is very scarce, and realistic data for multiple penetrants is totally absent from the open literature. Therefore, in the absence of a reliable data set, we will propose a simplified model accounting for 3 penetrants (i.e. quaternary system) in an LLDPE plant. Two systems are considered, at 90°𝐶, which are frequently employed in industry:

1. Copolymerization of ethylene and 1-hexene in presence of n-hexane as ICA 2. Copolymerization of ethylene and 1-butene in presence of iso-butane as ICA In the Sanchez-Lacombe EOS, the concentration of the different components in the polymer is predicted based on the binary interaction parameters 𝑘 between each pair of components (the penetrating species and the polymer). The thermodynamic model is based on the following assumptions: i) The gases dissolve only in the amorphous phase [24] ; ii) The interaction between molecules of olefins and/or ICA is ideal. [25] Thus, in a quaternary system of three penetrating gaseous species, ethylene (1), ICA (2) and comonomer (3), in the polymer (4), 𝑘 , 𝑘 and 𝑘 are equal to zero. Finally, only the global degree of crystallinity of the polymer is accounted for in the thermodynamic model. Strictly speaking, also tie molecules linking the crystalline lamellae can influence the solubility of different species in the amorphous phase. [26] Due to the lack of solubility data, the following, more critical assumptions are considered:

A.1 Additive effect: The quaternary system is approximated by a ternary system: ethylene/ (ICA+comonomer)/ LLDPE, as suggested by Alves et al. [21] Thus, a "pseudo" component, representing the mixture of ICA plus comonomer, is defined for which the thermodynamic parameters are identified. In this assumption, no interaction between ICA and comonomer is considered, which means that these species behave independently from each other as if they were present in a ternary system (PE, ethylene and either ICA or comonomer). This is not unreasonable if the comonomer and ICA are similar in structure. This assumption is thus applicable for the two systems studied in this work, i.e the comonomer 1-hexene and nhexane as ICA, as well as the comonomer 1-butene and iso-butane as ICA. However, this assumption does not mean that the ICA and the comonomer have the same solubility or cosolubility effect, as discussed in the following two sections.

A2. Polynomial approximations:

In order to reduce the computation time, the results from the SL EOS or the experimental results, are approximated by polynomials of degree 1 or 2, as suggested by Alves et al. [21] Different pathways were considered for the two systems investigated in the present work, as described in the following sections.

Polyethylene, in presence of ethylene, 1-hexene and the ICA n-hexane

For the first system, the comonomer 1-hexene and the ICA n-hexane, both the comonomer and the ICA were found to have comparable solubilities in a binary system, especially at low pressure, as shown by Figure 1 (Yao et al. [27] and Jin et al. [28] ). Therefore, Alizadeh et al. [23] assumed it safe to consider that they have similar solubility in LLDPE in a ternary or quaternary system. A similar observation was found for other 3 and 6 carbon pairs, such as the isomers propene and propane, where the difference in their solubility constant of Henry's law was 10% at 25°C, as reported by Michaels et al. [24] . This can be explained by the fact that 1-hexene and n-hexane have similar shapes and same number of carbons, so almost the same size, therefore, they have similar tendency to condense (i.e., same volatility). Besides, they have similar nature of interaction with LLDPE segments (i.e., same nature of van der Waals forces). [23] Therefore, in a quaternary system, the ratio of solubility of 1-hexene in LLDPE to n-hexane is assumed 𝑟 1. Note that for the second system, the comonomer 1-butene and the ICA iso-butane, their binary solubilities in LLDPE are different, and was accounted for as explained in section 2.1.2.

Figure 1. Solubility of n-hexane and 1-hexene in LLDPE (binary systems) at 70°C (data from Yao

et al. [27] and Jin et al. [28] )

For this system, copolymerization of ethylene with 1-hexene in presence of the ICA n-hexane, solubility data are available only for the ternary system ethylene/n-hexane/PE at 10 bar ethylene [27] (Figure 2). To use these data in a quaternary system (Assumption A1: Additive effect), the ICA n-hexane is assumed to thermodynamically behave like the comonomer 1hexene (as explained in the previous section). [23] The available ternary data of solubility (Figure 2) is then used to obtain linear or polynomial equations (Assumption A2: Polynomial approximations). It can be noticed that the solubility of ethylene varies linearly over a small range with the pressure of ICA/comonomer, therefore a polynomial of degree 1 could fit the experimental data, while the solubility of comonomer varies nonlinearly with its pressure, therefore a polynomial of degree 2 was necessary, as follows:

0 0.2 0.
Where M and 𝑃 are respectively the concentration in amorphous polymer (mol m -3 ) and pressure (bar) of component 𝑖 (1 for ethylene and 2 for the comonomer) and r represents the ratio of solubility of comonomer to ICA, which is equal to one in this system. The values of the coefficients (A, B, C, D) in equations 1 and 2 are given in Table 1. As the reaction takes place in the amorphous region only, the concentration of monomers in the amorphours phase is used to calculate the reaction rate. It is important to mention that these coefficients are valid for the specific conditions (temperature and pressure) for which they were developed (i.e. ethylene pressure of 10 bars and pseudo-component 'ICA+comonomer' pressure of 0 to 1 bar). It is also important to note that the concentration of ethylene only varies slightly as a function of the partial pressure of "n-hexane +1-hexene", however the concentration of n-hexane (and so of 1hexene) is very sensitive to the partial pressure of the pseudo-component 'ICA+comonomer', which is necessary to account for in the model (Figure 2). Note that when increasing temperature, the slope of the concentration of n-hexane with its partial pressure decreases (from C  624 to 506 mol m -3 Pa -2 ), while the slope of the concentration of ethylene increases (from A  34 to 279 mol m -3 Pa -1 ). Increasing the temperature thus increases slightly the impact of ICA on ethylene absorption, but it remains very low. Besides, the figure shows that the concentration of n-hexane in amorphous PE increases slightly in a binary system compared to a ternary system. This was expected given the anti-solvent effect of ethylene on ICA in a ternary system. [29,30] Note that the concentration plots presented in this paper were calculated by converting solubility data (in g g -1 amorphous polymer) found in the open literature into mol m -3 using the amorphous polymer densities (i.e., the values of the swollen polymer density with different ICAs) estimated by Sanchez-Lacombe EoS for each case. [31] and for binary systems from Yao et al. [27] )

Figure 2. Ternary solubility data ethylene/n-hexane/LLDPE at 90°C and 70°C at 10 bar of ethylene (experimental data are taken for ternary systems from Yao et al.

: a) Concentration of ethylene in amorphous LLDPE as a function of the partial pressure of n-hexane, b) Concentration of n-hexane as a function of its partial pressure. Table 1. Coefficients of the correlations allowing to estimate the ethylene and 1-hexene concentrations in amorphous LLDPE at 90 °C and 70 °C (valid at ethylene pressure of 10 bar and pseudo-component pressure on the range 0-1 bar)

Value at Equations 1 and 2, with their identified parameters in Table 1, allow for the estimation of the concentration of ethylene and 1-hexene in the quaternary system ethylene/1-hexene/n- hexane/LLDPE at equilibrium. This information is all what is required for the rest of the paper regarding this system, and there is no need for the SL EoS here.

90°C

Polyethylene, in presence of ethylene, 1-butene and iso-butane as ICA

For the copolymerization of ethylene with 1-butene in presence of iso-butane as ICA, ternary data is not available for either ethylene/iso-butane/LLDPE or for ethylene/1-butene/LLDPE. Moreover, the available binary sorption data (Figure 3) [32,33] indicates that 1-butene and isobutane have different solubilities in the polymer and cannot be assumed to be similar, as could be done for 1-butene/n-butane and 1-hexene/n-hexane. Indeed, the solubility of 1-butene in HDPE is higher than that of iso-butane by almost a factor of 𝑟 1.78, at nearly the same temperature. Due to this lack of data, the following assumptions are made only for this system:

A3. The ratio of sorption of 1-butene to iso-butane, 𝑟 (Figure 3), identified in a binary system, was assumed to remain unchanged in a ternary system, and to remain unchanged within a A4. The 𝑘 parameters are assumed to vary linearly with temperature. So, as solubility data is not available, an extrapolation of 𝑘 is done to predict them at 90°𝐶.

A5. The solubility of iso-butane in HDPE and LLDPE is assumed to be the same, as only solubility data in HDPE is available (in a binary system [32] ).

Again, Assumption A1 is applied to this system: The comonomer and ICA were assumed to have an additive effect in a quaternary system. Alves et al. [21] validated this assumption for propane and iso-butane based on reaction data from patents. This means that propane and isobutane do not affect the solubility of each other (i.e. there is no co-solubility effect), which is reasonable in view of their similarities. A similar assumption can be done regarding 1-butene and iso-butane in our system. Based on this assumption, Alves et al. [21] estimated the binary interaction parameters 𝑘 for the ternary system ethylene/iso-butane/LLDPE by fitting to experimental solubility data of the binary systems ethylene/LLDPE and iso-butane/HDPE at 70°C (Table 2). [32] Thus, the correlation between the ICA and the comonomer becomes (combining Assumptions A1, A2 and A3):

M 𝑟 𝑃 𝑃 𝑃 𝐸 𝑃 𝑃 (3)
In order to identify the parameters of equations 1 and 3 for the quaternary system, SL EOS is first used in the ternary system ethylene/iso-butane/LLDPE (with 𝑘 identified using binary solubility data [21] ) to identify the solubility of ethylene and iso-butane at different pressures of iso-butane. Then, the 𝑘 parameters were extrapolated over temperature to have values at 90°C (Assumption A4) (Figure 4, Table 3). The concentration of 1-butene is then calculated using 𝑆 𝑟 𝑆 . From the obtained data points, polynomials of order 1 were identified for both ternary systems.

As in the first system, it can be noted that the concentration of ethylene only varies slightly as a function of the partial pressure of the pseudo-component "isobutane+1-butene", while the concentration of comonomer 1-butene is very sensitive to its partial pressure (Figure 4). [21] 80°C 0 -0.022 0.022 (82°C) [21] 90°C 0 -0.032 -3.75 10 -4 T(K)+0.1551

Table 3. Coefficients of the correlations allowing to estimate the ethylene and 1-butene concentrations in amorphous LLDPE at 𝟗𝟎°𝑪 (valid at ethylene pressure of 7 bar and pseudocomponent pressure on the range 5-10 bar)

Parameter at 𝟗𝟎°𝑪 Units

Kinetic model

A classical reaction scheme (Table 4) of copolymerization of ethylene in presence of a catalyst having one type of active sites was considered (see for instance de Carvalho et al. [34] , McAuley et al. [35] , Chatzidoukas et al. [15] ). The only difference between the two copolymerization systems, with the comonomer 1-hexene or with 1-butene, is the values of the rate constants.

The following notations are used in Table 4: S : potential catalyst active sites, P : activated vacant catalyst sites, P * : total active sites (vacant P and occupied by a polymer chain P , ), P , :

living copolymer chains of length n ending with monomer i, D : dead copolymer chains of length n and C : deactivated catalyst sites.

The reaction rates resulting from the proposed reaction scheme are given in Table 5. All concentrations are in (mol m -3 ) and the following notations are used: λ moment 𝑘 of living chains, 𝜇 moment 𝑘 of dead chains and 𝑀 , is the cumulative polymer average molecular weight. The values of the different kinetic rate constants are given in Tables 6 and7. 𝐸 refers to the activation energy and k0 to the pre-exponential factor. For the case of ethylene-co-1-butene, the parameters were taken from Chatzidoukas et al. [15] or Ghasem et al. [37] . For the system ethyleneco-1-hexene in gas-phase, fewer parameters are available. Chakravarti et al. [38] gave some kinetic parameters for this system using a metallocene catalyst. In order to keep both systems comparable in terms of catalyst activity, only the reactivity ratios were taken from Chakravarti et al. [38] , and the other parameters and ratios were kept as for the first system. The identification of a specific kinetic model for a defined system is out of the scope of this work as our primary focus is to explore the impact of the co-solubility effect on grade change optimization. Parameter Value

Spontaneous activation [37] 𝑘 , s -1 7.210 𝑘 , m 3 mol -1 s -1 2.910 2 𝐸 , J mol -1 37656

Spontaneous chain transfer 15 𝑘 , m 3 mol -1 s -1 7.2

𝐸 J mol -1 33472
Transfer to hydrogen 37 𝑘 , m 3 mol -1 s -1 6.3

𝐸 J mol -1 33472

Transfer to monomer [37] 𝑘 , m 3 mol -1 s -1 0.15

𝑘 , m 3 mol -1 s -1 0.43 𝑘 , m 3 mol -1 s -1 0.15 𝑘 , m 3 mol -1 s -1 0.43 𝐸 , J mol -1 33472
Activation energy of propagation [15] 𝐸 J mol -1 37656

Table 7. Propagation rate coefficients of the co-polymerization of ethylene with a comonomer

1-Butene 1-Hexene

Reactivity ratios (-):

𝑟 𝑘 /𝑘 42.5 18.94 [38] 𝑟 𝑘 /𝑘 0.023 0.04 [38] Pre-exponential factor of propagation m 3 mol -1 s -1 : 𝑘 , 2.4810 4 [37] 2.4810 4 𝑘 , 5.8210 2 [37] 𝑘 /𝑟 1.310 3 𝑘 , 1.8610 4 [37] 𝑘 /𝑟 2.6510 3 𝑘 , 4.3710 2 1.0610 2 [38]*

Comonomer initiation m 3 mol -1 s -1 : 𝑘 , 40.7 [37] 𝑘 , 𝑘 /𝑘 1.25 𝐸 , J mol -1 37656 37656

* Calculated to respect the ratio 𝑘 /𝑘 in reference [38] .

Fluidized bed reactor model

The FBR is modeled as a single-phase CSTR. This assumption is a reasonable initial approximation in industrial FBR given its high recycle ratio and low single pass conversion. [8,35] Fresh or prepolymerized catalyst particles, gas species (monomers, N2, H2) and condensed gas (ICA) are assumed to be fed continuously at the bottom of the reactor. Thus, the FBR can roughly be divided into two compartments: one super-dry compartment in the containing only gas and polymer, and one much smaller compartment in the bottom also containing condensed vapors. Only the upper compartment is considered in the present model, which represents most of the reactor volume. Indeed, when operating under condensed mode, the injected liquid species evaporate rapidly and the major part of the reactor only contains solid and gas species (i.e. super-dry mode). [39] The mass balances of the different species in the FBR are given in Table 8, with the following

And the steady state mass balance for the polymer in the bed is given by the following equation: [40] 𝑄 𝑅 𝑀 , 𝑅 𝑀 , 𝑉 𝜌 𝐹 /𝜌 1 𝜀

where 𝜌 and 𝜌 are the densities of the polymer (around 920 kg m -3 ) and catalyst (2800 kg m -3 ), respectively. The dimensions of the bed are given in Table 9.

Correlations of key properties

The main properties to be controlled in the gas-phase copolymerization of ethylene are the melt index (𝑀𝐼, or melt flow index 𝑀𝐹𝐼, g/10 min) and the polymer density (𝜌 ). Correlations are therefore needed to estimate these properties. The available correlations in the literature can be divided into two categories:

A. Correlations which relate the final properties to the individual monomer conversions in the reactor (i.e. to reacted species) (Table 10). [11] These correlations are universal and remain valid when varying the operating conditions.

B. Correlations which relate the final properties to the operating conditions (i.e. T, P, or concentrations of unreacted species in the gas phase), such as [11] :

ln 𝑀𝐼 3.5 ln 𝑘 𝑘 𝑘 𝑘 𝑘 𝜌 𝜌 𝜌 ln 𝑀𝐼 𝜌 𝜌 (6)
Where 𝑘 , 𝜌 are tuning parameters and M and M are comonomers. Other correlations also exist in the open literature. [14] [41-44] Such correlations are only valid for the set of operating conditions for which they were derived and cannot be used to account for thermodynamic effects such as the co-solubility effect. Therefore, such correlations are not valid during grade transition where the operating conditions change.

In this work, the correlations of the first category will be employed (Table 10), as only such correlations would be able to account for the co-solubility effect for instance. In this work, the correlations developed by McAuley and MacGregor [11] for both MI and density are used (see Table 10).

The derivation of these correlations is based on physical interpretations. For instance, the melt index is highly correlated to the polymer molecular weight distribution and branching characteristics. [45] To simplify, the instantaneous melt index is usually correlated to the instantaneous average polymer molecular weight 𝑀 . The molecular weight is in turn affected by the concentration of monomer, comonomer and hydrogen. More particularly, hydrogen plays the role of a chain transfer agent in catalytic ethylene reactions, thus allowing to reduce the polymer molecular weight. [46] It constitutes therefore a primary manipulated variable during grade transitions. The melt index considered in this work represents the flow rate of a molten polymer under the load of 2.16 kg at 190 °C (ASTM D1238). [47,26] Concerning the polymer density, it is strongly affected by the length and the number of short chain branches. Hence, it is mainly governed by the fraction of reacted comonomer in the copolymer (𝐶 ) (Table 10). [48] By creating short chain branches on the polymer, the comonomer allows reducing the polymer density. As a consequence the crystallinity of the polymer also decreases. [26] The comonomer constitutes the second important manipulated variable during grade transition. The correlation proposed by McAuley and MacGregor [11] is based on patent data collected by Sinclair [START_REF] Sinclair | Process Economics Report[END_REF] , and it relates the instantaneous polymer density to the comonomer incorporation in the polymer by including 𝐶 (where 𝐶 𝜑 100, and 𝜑 , see Table 5).

Table 10. Correlations of Category A for the properties of the polymer: MI (the same relation is used for instantaneous and cumulative properties) and density

Ref

Melt index (g/10 min) Density 𝐠 𝐜𝐦 𝟑 Data from [11] 𝑀 kg mol 111525 𝑀𝐼 .

(or 𝑀𝐼 3.35 10 𝑀 . ) 𝜌 0.966 0.02386 𝐶 .

Butene grades [49] [50]

𝑀𝐼 2.7 10 𝑀 .

[51]

[51,51] 𝑀𝐼 4.195 10 𝑀 .

[48]

𝑀𝐼 3 10 𝑀 . 𝜌 0.023 ln 𝐶 0.9192 Octene grades [53] The cumulative properties (of the polymer exiting the reactor) can be calculated from the instantaneous ones (those being produced at time t) by integration over the residence time (𝜏).

Therefore, the cumulative melt index (𝑀𝐼 ) becomes: [START_REF] Lee | [END_REF] d 𝑀𝐼 . d𝑡

1 𝜏 𝑀𝐼 . 1 𝜏 𝑀𝐼 . (7)
Similarly, the cumulative density of the polymer (𝜌 ) is given by: [8] d 𝜌 d𝑡

1 𝜏 𝜌 1 𝜏 𝜌 (8)

Grade transition strategy

Dynamic optimization is employed to optimize the transition between different grades of LLDPE in the FBR using the combined kinetic and thermodynamic model.

Formulation of the optimization problem

The manipulated variables are the flow rates of hydrogen and comonomer and the controlled outputs are the melt index and the polymer density.

The optimization problem can be written as follows: [12] min 𝒖 𝐽 𝒖 𝑡 , 𝒙 𝑡 , 𝑡 ∈ 𝑡 , 𝑡

𝒖 𝒖 𝑡 𝒖 ( (9)
) 10
where J is the objective function, 𝒙 𝑡 is the vector of state variables (see Table 8) and 𝒖 𝑡 represents the vector of manipulated variables, 𝒖 𝑡 𝐹 , 𝐹 . The inequality constraints (10) indicate the available ranges of manipulated variables.

Objective function

The considered objective function is the following: [12] for instance. Takeda et al. [13] suggested that the choice between minimizing the transition product and the transition time should be based on the market demand: where at high polymer sales and plant capacity production it is preferred to minimize the transition time; while at low polymer sales and reduced plant capacity it is preferred to minimize the transition production and authorize a longer transition time. A transition product can usually be sold, although at a discounted price.

𝐽 𝒖 𝑤 𝑤 𝑤 𝑤 d𝑡 𝑤 ∑ (

Degrees of freedom of the inputs

It is usually sufficient to assume the manipulated variables (here, the flow rates of hydrogen and comonomer) to vary by a series of ramps during the transition. [12] Based on the literature study and the residence time of the FBR (4-6 hours in this study, depending on the operating conditions), the transition is divided into 5 intervals, where the final interval corresponds to the steady state interval, to maintain until the end of the production of the new grade. Thus, the optimization allows switching the flow rates every 2 hours during the first 8 hours, and the last ramp corresponds to the steady state flow rate of the new grade.

Simulation results and discussion

The proposed strategy is evaluated in grade transition starting from grade A, to grade B with higher or lower MI and , then coming back to grade A, for both of the copolymerization systems (Table 11). These choices are based on LLDPE specifications, i.e. MI[0.01-100] g/10min [55] and [915-935] kg m -3 [56] . The duration of each grade production is usually defined by the market demand, the specifications or the claims of the production. Here, an arbitrary duration of production of each grade of 30 hours is implemented in both systems. No particular change is required at the optimization level to change to shorter or longer production periods, only the final times need to be indicated. The initial steady state conditions, producing grade A, are given in Table 12. This weighting factors were tuned as follows, except otherwise mentioned: w1 = 0.08, w2 = 1, w3 = 8 w4 = 19 and w5 = 10 4 . This choice was based on few simulation tests, in a way to ensure a compromise between fast convergence of the cumulative properties while reducing the overshoots of the instantaneous properties. Indeed, while allowing for big variations in the instantaneous properties leads to a faster convergence of the cumulative properties, some conditions of comonomer or hydrogen pressures might lead to polymer softening or sticking problems. [START_REF] Mckenna | [END_REF] In the following simulations, temperature control is assumed to be perfect in the bed, so that the working temperature is constant.

Copolymerization of ethylene and 1-hexene in presence of n-hexane as ICA

The optimization strategy was evaluated using the parameters of the first system, i.e. the copolymerization of ethylene with 1-hexene in presence of n-hexane. Note that the thermodynamic model was developed for this system for ethylene pressure of 10 bar and pseudo-component (i.e. comonomer plus ICA) pressure on the range of 0-1 bar. Therefore, the simulations (including the choices of the set-points) are conducted in a way to respect these ranges.

Figure 5 shows the results of the two grade transitions, from A to B, and from B back to A. This scenario was simulated using the following weighting factors: w1 = 0.08, w2 = 1, w3 = 8 w4 = 19 and w5 =0, therefore the instantaneous properties (𝑀𝐼 and 𝜌 ) go beyond the set-point (SP)

during the transition in order to accelerate the convergence of the cumulative properties (𝑀𝐼 and 𝜌 ). This is related to the variations of the flow rates of hydrogen and 1-hexene, which are higher at the beginning of the grade transition and then they stabilize, as indicated by the increase in the pressure. The overshoots in the instantaneous properties can be reduced by reducing the weighting factors multiplying the cumulative properties w2 and w4 compared to those of the instantaneous properties w1 and w3, or by considering w5  0, or by adding constraints on the outputs, as discussed in the following scenarios. The MI is inversely proportional to the polymer molecular weight. Therefore, an increase in the hydrogen pressure during the transition, from grade A to B for instance, led to a decrease in the polymer molecular weight and thus to an increase in the melt index. Likewise, an increase in the comonomer pressure during the transition from grade A to B, led to an increase in the amount of short branches and thus to a decrease in the polymer density. The proposed strategy allows to move either to higher (grade A to grade B) or lower (B to A) values of 𝑀𝐼, and vice versa for 𝜌. The figure shows that the concentration of ethylene in the polymer particles (which constitutes the site of the reaction) does not change significantly during the transition, where the comonomer flow rate is varied, so the co-solubility effect is negligible in this sense and under the realized changes in the comonomer pressure. Note that the ethylene pressure is maintained constant in all the grades. However, the concentration of comonomer in the polymer particles is highly affected by these changes, which demonstrates the necessity of using a good thermodynamic model. The impact of the thermodynamic model is investigated more deeply in the last section. Note that the total pressure of comonomer and ICA reached 1.35 bar at the maximum in this simulation, but only for a short duration, and therefore the employed thermodynamic correlation remains valid during most of the time (PICA+Pcom=0-1 bar). )

Figure 6. Grade transition in ethylene-1-hexene co-polymerization in presence of n-hexane: Tracking of the cumulative properties (w 1 =w 3 =w 5 =0 ; w 2 =1 ; w 4 =4), with constraints on the instantaneous properties (0.1<MI i <3 and 910< i <935).

The same scenario presented in Figure 5Figure 7. Grade transition in ethylene-1-butene copolymerization in presence of iso-butane (w1 = 0.08, w2=1, w3=8, w4 = 19 and w5 = 0). was simulated while tracking only the cumulative properties (i.e. w1=w3=0) and considering constraints on the instantaneous as follows: 0.1<MIi<3 and 910<i<935 (Figure 6).

It can be seen that the convergence of the cumulative properties is slowed down compared to The proposed grade transition optimization strategy was evaluated in the second system: ethylene and 1-butene copolymerization in presence of iso-butane as ICA. Note that the thermodynamic model was developed for different conditions for this system: i.e. ethylene pressure of 7 bar and pseudo-component pressure on the range of 5-10 bars. The set-points of the melt index and polymer density of grades A and B were also set differently in this system, but still within LLDPE grades. The same factors as the first system were considered. The last term of the objective function (𝑤 ∑ ) can allow minimizing the variation of the manipulated variables (flow rates of hydrogen and the comonomer), and thus to reduce the overshoots in the instantaneous properties. Indeed, injecting big amounts of hydrogen or rapidly increases the risk of polymer softening and stickiness. [START_REF] Mckenna | [END_REF] As a consequence, adding this term is expected to reduce the overshoots in instantaneous properties. Due to the low values of the variations of the flow rates, it was necessary to have a high weighting factor, w 10 , to ensure an impact on the performance. Figure 8 shows the results when adding this term to the objective function, which is to be compared to Figure 7 done under the same conditions but without this term. The figure clearly shows that the pressures of hydrogen and comonomer undergo less changes. As a consequence, the instantaneous properties have lower overshoots. However, this delays a little the convergence of the cumulative properties to the set-points. A compromise is thus to be determined between fast convergence of the cumulative properties and less variation in the instantaneous properties. In order to evaluate the gain realized by the optimization strategy, its performance was compared to the case of injecting the optimal feed rates of the final grade during the transition (here called the final steady-state, SS), as done for instance by Rahimpour et al. [8] (Figure 9).

When employing a constant flow rate during the transition, the convergence time is that of the residence time of the reactor. It can be seen that employing the optimized varying flow rates during the transition allows to reduce the convergence times of the cumulative melt index and the density.

. Grade transition in ethylene-1-butene copolymerization: comparison between the proposed grade transition strategy (leading to varying optimized flow rates during the transition) and injecting a constant flow rate during the transition (corresponding to the optimal flow rate of the final grade, under steady-state conditions)

Impact of the thermodynamic model

In order to demonstrate the importance of employing a good thermodynamic model in the optimization strategy, two scenarios were simulated. The first scenario was performed by assuming an error in the parameters of the thermodynamic model. The second system was used for this purpose, i.e. ethylene-1-butene co-polymerization in presence of iso-butane as ICA.

In Figure 10, an error is assumed in the parameters A and E in equations 1 and 2, related to the calculation of ethylene and comonomer concentrations in polymer, M and M , respectively. It can be seen that the employed flow rates bring the process to different set-points than the desired ones (lower MI and , as the used parameters A and E were assumed to be underestimated). Indeed, using lower A and E parameters in the model gives lower M and M than the real ones. In order to correct ratios of monomer to hydrogen as well as to comonomer (to obtain the desired properties), the optimization strategy forces the decrease in the hydrogen flow rate, which leads to an increase in the polymer molecular weight, and a decrease in the MI. Similarly, the optimization forces the decrease in the comonomer flow rate, and as a result of errors in M and M , a decrease in the polymer density is observed in this case). Note that the optimization strategy continues to work adequately, but as it is based on a wrong model it does not converge to the correct optimal points, therefore the use of an adequate thermodynamic model is essential. The second scenario was performed by switching to a binary model to describe the solubility of the different species in the polymer (i.e. with no co-solubility effect). The system ethylene-1-hexene co-polymerization in presence of n-hexane as ICA was used for this simulation. In this case, the thermodynamic model leads to the calculation of an incorrect concentration of ethylene in the amorphous phase of the polymer, M =257 mol m -3 at 10 bar ethyleneand 90°C

(as it assumes a binary system) [27] instead of around 280 mol m -3 estimated in the pseudoquaternary system. It also calculates an incorrect comonomer concentration in the polymer particle M . The concentration of 1-hexene in a binary system (1-hexene+LLDPE) [29] is expected to be higher compared to its concentration in a ternary system due to the anti-solvent effect of ethylene (as shown in Figure 2). However, combining ICA+comonomer in a pseudoquaternary system leads to a global pressure which is much higher. As indicated by Figure 4, a small change in the pressure leads to a high change in the solubility of the pseudo-component, so the quaternary system leads to a much higher concentration of comonomer in the particles than in a binary system at the same pressure. Note however that the same flow rate is injected in the model and the process, but different reaction rates occur (due to the use of different thermodynamic models), therefore the comonomer pressure varies a lot between the two simulations, and therefore it is not straightforward to compare the concentration of comonomer in the model and the process in this simulation. In this simulation, when M = 165 mol m -3 in the pseudo-quaternary system, it was M = 159 mol m -3 in the binary model.

The simulation test was performed using the binary model for both the concentrations of ethylene and 1-hexene in the amorphous phase of the polymer (so the model and grade transition is simulated using the binary model while the process is simulated using the pseudoquaternary model). Figure 11 shows that using a binary thermodynamic model and ignoring the co-solubility effect leads to a big drift of the properties from the set-points. Indeed, the model assumes a lower M (so a lower polymer molecular weight and a higher MI). Therefore, the optimization strategy based on this model makes the decision to decrease the hydrogen flow rate. However, when implemented to the process (simulated using the pseudo-quaternary model, where the concentration of monomer is higher), this flow rate leads to a higher MW, so to a lower MI. Following the same reasoning, a drift in the polymer density occurs due to errors in both M and M . This simulation demonstrates the importance of using an adequate thermodynamic model in the optimization strategy.

Conclusions

In this work, off-line dynamic optimization was implemented to optimize the grade transitions in a fluidized bed reactor of polyethylene. A combined kinetic and thermodynamic model was used in order to account for the co-solubility effects of the different gas species. The thermodynamic model is based on Sanchez-Lacombe EoS, but then simplified correlations were used to reduce the calculation time. Two copolymerizations were considered, the copolymerization of ethylene with 1-hexene in presence of n-hexane as ICA and the copolymerization of ethylene with 1-butene in presence of iso-butane. Some assumptions were made, mainly due to the lack of thermodynamic data in the literature, to allow the prediction of the solubility of the different species in PE in these quaternary systems.

The simulation results demonstrate the importance of the thermodynamic model in the optimization strategy. A good control of the polymer melt index and density could be realized by manipulating the flow rates of hydrogen and comonomer. Nevertheless, in both systems, the co-solublity effect of comonomer on ethylene was not observed, which is due to the low impact of the pseudo-component on the solubility of ethylene under the employed operating conditions Both the instantaneous and the cumulative properties could be controlled, in a duration much lower than the residence time of the reactor. The role of the weighting factors, in the minimization function, is determinant at this level, where it can either give more importance to controlling the instantaneous properties (thus eliminating any overshoot) or on the contrary allow a faster control of the cumulative properties in detriment of the instantaneous ones. A compromise between these two options is necessary in order to ensure a fast convergence of the cumulative properties to the set-points (thus reduce the transition product) while avoiding big variations in the flow rates or pressures of hydrogen and comonomer as they may increase the risk of polymer sticking or softening. To reach the same objective, constraints on the instantaneous properties can be considered, but this slows down the calculation.

The proposed optimization tool should allow a more efficient operation and a better control of the polymer quality. The kinetic parameters and the correlations of the polymer properties used in this work were taken from literature, as well as the assumption of the bed to behave as a CSTR. These models can be replaced by more detailed models when available in the same optimization strategy. For instance, improvement at the level of the bed model can be done by considering a compartmental model and at the kinetic level by considering multiple site catalysts leading to bimodal molecular weight distributions and using correlations of the MI and polymer density adapted to such systems. Finally, the availability of more thermodynamic data or the use of a particle model accounting for diffusion would allow to improve the precision of the optimization strategy outcome.

Figure 3 .

 3 Figure 3. Binary solubility data in HDPE of a) comonomer 1-butene at 𝟔𝟖. 𝟗°𝑪, and b) ICA isobutane at 65.6°C.

  range of 65-90 °C. This means that they are assumed to have the same cosolubility effect on ethylene.

Figure 4 .

 4 Figure 4. Concentrations in amorphous LLDPE obtained using SL EoS, Pseudo-quaternary system ethylene/(ICA+comonomer)/HDPE, at 𝟗𝟎°𝑪 (after extrapolation of 𝒌 𝒊𝒋 ) and 7 bar of ethylene: a) ethylene and b) comonomer 1-butene.

  notations, 𝜀 : porosity of the bed, 𝑋 and 𝑋 , : mass fractions of component i in the recycle and the input stream respectively, ℎ(m): bed height, 𝑉 m : bed volume, 𝐴 m : crosssectional area of the bed 𝑄 (m 3 s -1 ): bed outlet volumetric flow rate, 𝐹 (kg s -1 ) inlet flow rate of component 𝑘, 𝐹 (kg s -1 ) recycling flow rate and 𝑋 mass fraction of component 𝑘 in the gas phase.

11 )

 11 By considering both instantaneous and cumulative properties (of the melt index and polymer density), and by a good tuning the weighting factors 𝑤 (with 𝑖 1 5), one may accelerate the convergence of cumulative properties while keeping the instantaneous properties within an acceptable range. The last term on the right hand side of this equation is also intended to minimize the variation of the input during the transition, in order to avoid oscillations (as in model predictive control). The normalization of the different terms (i.e. the division by the setpoints of the MI and density) allows for an easier tuning of the weighting factors. The indices i, c and sp refer to the instantaneous, cumulative and set-point properties, respectively. At a constant reaction rate, the proposed objective function allows reducing the quantity of transition product as well as the transition time, even though the time is not explicitly minimized in this function. If the reaction rate varies during the transition, then this objective function allows minimizing only the transition time. The minimization of the transition product would necessitate, in case of variable reaction rate, to multiply the criterion by the instantaneous reaction rate (Rp), as done by McAuley and MacGregor

Figure 5 .

 5 Figure 5.Grade transition in ethylene-1-hexene copolymerization in presence of n-hexane at 90°C(w 1 = 0.08, w 2 =1 ; w 3 =8, w 4 = 19, w 5 = 0).

Figure 5 ,

 5 Figure5, but the overshoots in the instantaneous properties (𝑀𝐼 and 𝜌 ) are reduced and kept within the constraints. Adding hard constraints allows remaining within the acceptable range of properties, but it slows down the calculation. Another way to reduce the overshoots in the instantaneous properties, without considering constraints, consists of increasing the values of w2 and w4 with respect to w1 and w3 or by considering w5  0.

2

 2 Copolymerization of ethylene and 1-butene in presence of iso-butane as ICA

Figure 7 .

 7 Figure 7. Grade transition in ethylene-1-butene co-polymerization in presence of iso-butane (w 1 = 0.08, w 2 =1, w 3 =8, w 4 = 19 and w 5 = 0).

Figure 7

 7 Figure7shows the simulation results. The melt index converges in about 6 hours to the setpoint, while the density converges to the set-point in 2 hours. The overshoots of the instantaneous properties remain acceptable, but they can be reduced by either manipulating the weighting coefficients (as discussed in the following scenario) or by adding hard constraints on

Figure 8 .

 8 Figure 8. Grade transition in ethylene-1-butene: effect of the term 𝒘 𝟓 ∑

Figure 9

 9 Figure 9. Grade transition in ethylene-1-butene copolymerization: comparison between the proposed grade transition strategy (leading to varying optimized flow rates during the transition) and injecting a constant flow rate during the transition (corresponding to the optimal flow rate of the final grade, under steady-state conditions)

Figure 10 .

 10 Figure 10. Influence of the thermodynamic model parameters on the process response, in ethylene and 1-butene copolymerization (Process parameters: A =1.98 mol m -3 bar -1 , E=180 mol m -3 , Modelparameters used for optimization: A =0.992 mol m -3 bar -1 , E=90.2 mol m -3 )

Figure 11 .

 11 Figure 11. Influence of using binary thermodynamic model (not taking in account the co-solubility effect) on the process response. System ethylene-1-hexene copolymerization in presence ofnhexane at 90°C. (w 1 = 0.08, w 2 =1, w 3 =8, w 4 = 19 and w 5 = 10 4 )

  temperature). The importance of the thermodynamic model was mainly related to evaluating the concentration of comonomer in the polymer during the transition, which highly impacts the polymer properties.

Table 2 . Binary interaction parameters of the ternary system ethylene/iso-butane/LLDPE (based on binary thermodynamic data)

 2

	Temperature °𝐶	Ethylene/iso-	Ethylene/LLDPE	Iso-butane/LLDPE
		butane 𝑘	𝑘	23	𝑘
	70°C	0	-0.014	0.025 (74°C)

Table 4 . Kinetic scheme of the copolymerization of ethylene with a catalyst of one site (without co- catalyst)

 4

	Designation		Reaction
	Spontaneous activation		S → P
	Chain initiation	P	M	P ,
	Propagation	P ,	M ⎯ P ,

Table 5 . Reaction rates of the different species (mol m -3 s -1 )

 5

	𝑅	𝑘 S
	𝑅	𝑘 S	𝑘	P		𝑘 M	𝑘 M	P		𝑘	𝑘 H 𝜆
	𝑅	P 𝑘 M	𝑘 M	𝜆 𝑘	𝑘	𝑘 H
	𝑅	P 𝑘 M	𝑘 M	𝜆	𝑘	𝜙	𝑘	𝜙	𝑘 𝜙
	𝑘 𝜙 M	𝑘	𝜙	𝑘	𝜙	𝑘 𝜙	𝑘 𝜙 M	𝜆 𝑘	𝑘
	𝑘 H	𝑘	𝜙	𝑘	𝜙 M	𝑘	𝜙	𝑘	𝜙 M
	𝑅	P 𝑘 M	𝑘 M	λ	𝑘	𝜙	𝑘	𝜙 M	𝑘	𝜙
	𝑘	𝜙 M		λ	2λ		𝑘 𝜙	𝑘 𝜙 M	𝑘 𝜙	𝑘 𝜙 M
	𝜆 𝑘	𝑘	𝑘 H	𝑘	𝜙	k	𝜙 M	𝑘	𝜙	𝑘	𝜙 M
	𝑅	λ 𝑘	𝑘	𝑘 H	𝑘	𝜙	𝑘	𝜙 M	𝑘	𝜙
	𝑘	𝜙 M
	𝑅	λ 𝑘	𝑘	𝑘 H	𝑘	𝜙	𝑘	𝜙 M	𝑘	𝜙
	𝑘	𝜙 M
	𝑅	λ 𝑘	𝑘	𝑘 H	𝑘	𝜙	𝑘	𝜙 M	𝑘	𝜙
	𝑘	𝜙 M
	𝑅	𝑘 H 𝜆
	𝑅	𝑘 P	𝑘	𝜙	𝑘	𝜙	𝑘 𝜙	𝑘 𝜙 𝜆 M
	𝑅	𝑘 P	𝑘	𝜙	𝑘	𝜙	𝑘 𝜙	𝑘 𝜙 𝜆 M
	With 𝜙				, 𝜙	1 𝜙 ,

H (mol m -3 am. polymer)=𝑆 , 𝑆 (g H2 g -1 am. polymer)=10 -9 𝑃

[36]

Table 6 . Pre-exponential factors and activation energies of the kinetic parameters of co- polymerization of ethylene and a comonomer (common values for both systems) (𝒌 𝒊

 6

𝒌 𝒊,𝟎 𝒆 𝑬 𝒂 𝑹𝑻 )

Table 8 : Mass balances of the different species in the FBR

 8

	Monomer i	d M d𝑡	𝐹 𝑀 𝜀 𝑉	𝑄 M 𝑉	1 𝜀 𝜀	𝑅	M 𝐴 𝑉	dℎ d𝑡
	Hydrogen	d H d𝑡	𝐹 𝑀 , 𝜀 𝑉		𝑄 H 𝑉	1 𝜀 𝜀		𝑅	H 𝐴 𝑉	dℎ d𝑡
	Nitrogen	d N		𝐹		𝑄 N	N 𝐴	dℎ
		d𝑡	𝑀 , 𝜀 𝑉		𝑉	𝑉	d𝑡
	ICA	d ICA		𝐹		𝑄 ICA	ICA 𝐴	dℎ
		d𝑡		𝑀 , 𝜀 𝑉	𝑉	𝑉		d𝑡
	Potential catalyst sites S	d S d𝑡	𝐹 S , 𝜌 1 𝜀	𝑉	𝑄 S 𝑉	𝑅		S 𝐴 𝑉	dℎ d𝑡
	Y: P , λ , λ , λ , 𝜇 , 𝜇 , 𝜇	d Y d𝑡	𝑅	𝑄 Y 𝑉	Y 𝐴 𝑉	dℎ d𝑡

Table 9 . Reactor dimensions

 9

	Parameter	Designation	Value
	Dbed	Bed diameter	4.75 m
	𝜀	Bed voidage	0.7
	h	Height of the bed	13.3 m

Table 11 . LLDPE grades considered in the grade transition policy

 11

	Grade	Melt index Target (g / 10 min) 1-hexene 1-butene	Density Target (kg m -3 ) 1-hexene 1-butene
	A	0.54	4.5	923	918.5
	B	2	2	916	922
	A	0.54	4.5	923	918.5

Table 12 . Initial conditions of the grade transition simulations (leading to grade A under steady state)

 12

		1-hexene	1-butene
	T (°C)	90	90
	𝑃 (bar)	9.4	7
	𝑃 (bar)	0.35	1.55
	PICA (bar)	0.6	3.5
	𝑃 (bar)	2.2	2.2

Acknowledgements

This work was funded by Agence Nationale de la Recherche (Thermopoly project, grant N° ANR-16-CE93-0001-01). The authors would like to thank M. Robert Pelletier for his input on grade transitions in polymerization processes.


Matthieu Jamme
email: matthieu.jamme@ght-yvelinesnord.fr

Leopold Oliver

Julien Ternacle

Raphael Lepeule

Amina Moussafeur

Jean-Philippe Haymann

Sovannarith San

Antonio Fiore

Nicolas Mongardon

Michel Daudon

Pascal Lim

Emmanuel Letavernier

Amoxicillin crystalluria is associated with acute kidney injury in patients treated for acute infective endocarditis

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Acute kidney injury (AKI) is a common complication of infective endocarditis (IE)

with a reported prevalence of 30-50% [START_REF] Majumdar | Renal pathological findings in infective endocarditis[END_REF][START_REF] Tamura | Long-Term Outcome of Active Infective Endocarditis with Renal Insufficiency in Cardiac Surgery[END_REF] . Antibiotic-associated crystals (AAC) nephropathy has been increasingly described as a cause of severe AKI secondary to antibiotic use [START_REF] Vodovar | Dramatic Increase of Amoxicillin-Induced Crystal Nephropathy Found in a Cohort Study of French Pharmacovigilance Centers[END_REF][START_REF] Garnier | High Incidence of Amoxicillin-Induced Crystal Nephropathy in Patients Receiving High Dose of Intravenous Amoxicillin[END_REF] . Amoxicillin crystals are frequently observed but sulfamethoxazole, sulfadiazine, ceftriaxone and other antibiotics may form AAC in urine [START_REF] Daudon | Drug-Induced Kidney Stones and Crystalline Nephropathy: Pathophysiology, Prevention and Treatment[END_REF] . However, the prevalence of AAC in the urine of IE patients is unknown. Moreover, the impact of crystals on the renal function is unknown and always retrospectively described after exclusion of differential diagnosis [START_REF] Vodovar | Dramatic Increase of Amoxicillin-Induced Crystal Nephropathy Found in a Cohort Study of French Pharmacovigilance Centers[END_REF][START_REF] Garnier | High Incidence of Amoxicillin-Induced Crystal Nephropathy in Patients Receiving High Dose of Intravenous Amoxicillin[END_REF] .

We aimed at (i) assess the prevalence and type of AAC in the urine of patients treated for suspected IE, (ii) identify factors associated with AAC and (iii) assess renal prognosis of AAC-positive patients.

We performed crystalluria study in adults patients admitted for suspicion of IE between 2017, June, 1 st and 2018, June, 1 st . All crystalluria studies were realized in fresh morning urine samples collected, transferred and analyzed within 4 hours. Downloaded from https://academic.oup.com/ndt/advance-article/doi/10.1093/ndt/gfab074/6225010 by chi poissy st germain, matthieu.jamme@ght-yvelinesnord.fr on 15 April 2021 Crystalluria examinations were performed without knowledge of the clinical status and biological results of the patients. Crystalluria was considered as positive if antibiotic crystals were observed in urine.

Primary outcome was the occurrence of urine AAC. Secondary outcome was the occurrence of AKI stage ≥ 2 according to the KDIGO classification.

To identify risk factors of AAC, because several crystalluria analyses were sometimes performed in a same patient, we used a mixed multivariate logistic regression. To handle with this competing risk setting and time-dependent covariates, we used a Cox time-dependent cause specific model [START_REF] Latouche | A note on including time-dependent covariate in regression model for competing risks data[END_REF] . Multivariate model was built using all variables with a P value lower than 0.20 in univariate analysis and then a stepwise backward selection was applied to select final variables. All statistical tests were two-sided using a type I error of 0.05. This study has been approved by the ethics committee of the French Intensive Care Society (ref. CE SRLF, #24-01).

Sixty-five crystalluria studies were performed in 34 patients with a median of 2 [1-2] samples per patient. Most of patients were men (71%) and the median age was 70 [62-77] years. Baseline characteristics are shown in Supplemental Table 1.

First-line antibiotics were amoxicillin (88%), gentamycin (82%) and cefazolin (29%).

After adaptation to the pathogen, second-line antibiotics were amoxicillin (60%), ceftriaxone (20%), vancomycin (20%) and rifampicin (20 %). Twenty-one (62%) patients underwent surgery within a median time of 3 [2-8] days. Supplemental Table 2 reports treatments and events that occurred during hospitalization.

Crystalluria studies were realized within a median time of 3 [2-5] days after diagnosis. AAC was observed in 18 (28 %) urinary samples from 14/34 (41 %) patients. All AAC identified were amoxicillin crystals (Figure 1). When positive, the median number of crystals was 8 [3-18]/mm 3 and a microscopic hematuria was associated in 13/18 (72 %) of AAC cases. We identified 6 other positive crystalluria with uric acid (n = 2) or calcium oxalate crystals (n = 4): 4/47 (8.5%) in the No AAC group and 2/18 (11.1%) in the AAC group. Median serum creatinine level at the time of crystalluria was 99 [65-Downloaded from https://academic.oup.com/ndt/advance-article/doi/10.1093/ndt/gfab074/6225010 by chi poissy st germain, matthieu.jamme@ght-yvelinesnord.fr on 15 April 2021 202] µmol/L. Multivariate logistic mixed effect analysis identified blood amoxicillin concentration (OR = 1.03 [1.01-1.05] per increase of 1mg/L), urine pH (OR = 0.78 [0.62-0.97] per increase of 0,1 unit) and a trend was observed with urinary specific gravity (OR for = 1.25 [0.80-2.73] per increase of 10 unit, p=0.10) as risk factors of AAC (Supplemental Table 3). AKI occurred in 16 (47%) patients. The median time from admission to AKI was 5 [3-7] 1).

This study is the first to assess the prevalence and prognosis of AAC in a context of IE by repeated analysis of crystalluria.

In 2020, Garnier et al. identified, among 358 patients treated with high daily dose of amoxicillin, 73 (20%) patients with AKI. AKI seemed related to antibiotic-induced crystal nephropathy in 16 patients. However, crystalluria study was performed 4.5 days after the onset of AKI and was positive in only 2 patients [START_REF] Garnier | High Incidence of Amoxicillin-Induced Crystal Nephropathy in Patients Receiving High Dose of Intravenous Amoxicillin[END_REF] .

In the present work, we highlight that high blood concentration of amoxicillin, low urinary pH and urine density are risk factors for amoxicillin crystal formation in urine.

Crystals may form in urine when antibiotics are in a supersaturation state, depending on antibiotic concentration in urine and pH [START_REF] Vodovar | Dramatic Increase of Amoxicillin-Induced Crystal Nephropathy Found in a Cohort Study of French Pharmacovigilance Centers[END_REF][START_REF] Sjovall | Renal excretion of intravenously infused amoxycillin and ampicillin[END_REF] .

Finally, we identify that AKI is a common complication of IE, and the presence of AAC in urine is an independent risk factor associated with the occurrence of renal dysfunction. Our prevalence of AKI is quite similar to those already reported after IE 8,9 .

In addition to amoxicillin crystals, we identified two other independent risk factors of AKI: use of vancomycin, a potentially nephrotoxic drug, and septic shock at admission 10,11 .

Our study has strengths. First, we used polarized microscopy to detect crystals.

Second, we prospectively analyzed the patients admitted in our center, not only patients with renal failure, to not overestimate the consequences of AAC. Third, because of the dynamic process of crystal formation, we collected several urine samples per patient to not underestimate the prevalence of AAC. Fourth, our biostatistical model is the most appropriate to handle with concurrent risk context and time-dependent covariate.

However our study has also major limitations. First, the small number of patients limits the power of statistical tests. Second, we could not apply a standardized chronological protocol of urine collection because crystalluria was not available during weekend and night. Third, no renal biopsy was performed in patients with AKI.

In conclusion, amoxicillin crystals are frequent in patients treated with high dose of amoxicillin for suspected IE, and their presence in urine predicts AKI. Our results suggest that repeated crystalluria analyses should be performed in patients receiving high doses of amoxicillin. An increase in urine output, urine alkalinization and a decrease in amoxicillin dose or a switch to other antibiotics may prevent AKI in these patients, deserving further studies.

AUTHORS' CONTRIBUTIONS

MJ designed the study, collected the data, performed the statistical analysis and drafted the manuscript. EL designed the study, performed crystalluria and drafted the manuscript. LO designed the study, collected the data and drafted the manuscript. JT, RL, AM, JPH, SS, AF, NM, MD and PL collected the data, contributed to data interpretation and analysis, and revised the manuscript for important intellectual content.

CONFLICT OF INTEREST STATEMENT

Downloaded from https://academic.oup.com/ndt/advance-article/doi/10.1093/ndt/gfab074/6225010 by chi poissy st germain, matthieu.jamme@ght-yvelinesnord.fr on 15 April 2021 None declared. This work was presented at the fourth congress of the French society of nephrology -dialysis -transplantation (SFNDT), 1-4 October 2019, Nancy, France and at the congress "Réanimation 2020", 5-7 February 2020, Paris, France.

TABLES

  8. Legrand M, Pirracchio R, Rosa A, et al. Incidence, risk factors and prediction of post-operative acute kidney injury following cardiac surgery for active infective endocarditis: an observational study. Crit Care 2013;17(5):R220. 9. Gagneux-Brunon A, Pouvaret A, Maillard N, et al. Acute kidney injury in infective endocarditis: A retrospective analysis. Med Mal Infect 2019;49(7):527-33. 10. Luque Y, Louis K, Jouanneau C, et al. Vancomycin-Associated Cast Nephropathy. J Am Soc Nephrol 2017;28(6):1723-8. 11. Hoste EAJ, Bagshaw SM, Bellomo R, et al. Epidemiology of acute kidney injury in critically ill patients: the multinational AKI-EPI study. Intensive Care Med 2015;41(8):1411-23.

  days. Multivariate Cox cause specific time-dependent analysis identified AAC (CSH = 7.44 [1.83-30.25]), septic shock at admission (CSH = 13.29 [2.25-78.60]) and use of vancomycin in first-line therapy (CSH = 5.49 [1.07-28.22]) as risk factors of AKI occurrence (Table

Table 1 . Time-dependent survival competing risk analysis of AKI

 1 -specific hazard; CI, confidence interval; BMI, body mass index; CKD, chronic kidney disease; IE, infective endocarditis; WBC, white blood cells; SCr, serum creatinine; AAC, antibiotic associated crystals in urine; AST, aspartate aminotransferase; ALT, alanine aminotransferase Initial multivariate model before stepwise variable selection included following covariates: Septic shock at admission, Streptococcus group, SCr at admission, 1st line Vancomycin, AST and ALT at admission, Positive AAC and surgery

		Univariate Analysis	Multivariate Analysis
		CSH	95% CI	P	CSH	95% CI	P
	Age > 70 years	0.75 0.26-2.17 0.59
	Male gender	0.85 0.28-2.53 0.76
	BMI (+1 kg/m 2 )	0.98 0.87-1.10 0.72
	History of valvulopathy	1.10 0.24-4.93 0.90
	History of arterial hypertension 0.67 0.23-1.93 0.46
	History of mellitus diabetes	0.62 0.14-2.75 0.52
	History of CKD	0.57 0.10-4.36 0.58
	History of solid cancer	0.72 0.16-3.34 0.70
	Septic shock at admission	8.94	1.59-50.17	0.01 13.29	2.25-78.60	0.004
	Congestive signs at admission 0.38 0.10-1.66 0.19
	Fever	1.26 0.35-4.54 0.72
	Native valve IE	1.25 0.44-3.57 0.68

* These variables had been analyzed as time-dependent covariates

Downloaded from https://academic.oup.com/ndt/advance-article/doi/10.1093/ndt/gfab074/6225010 by chi poissy st germain, matthieu.jamme@ght-yvelinesnord.fr on 15 April 2021

FUNDING

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

SUPPLEMENTAL MATERIALS

Supplemental Table 1. Baseline characteristics Supplemental Table 2. Treatment and evolution during hospitalization Supplemental Table 3. Comparison according to occurrence of AAC in urine Downloaded from https://academic.oup.com/ndt/advance-article/doi/10.1093/ndt/gfab074/6225010 by chi poissy st germain, matthieu.jamme@ght-yvelinesnord.fr on 15 April 2021


Rotational diode: Clockwise/counterclockwise asymmetry in conducting and mechanical properties of rotating (semi)conductors

Maxim N.

Rotational diode: Clockwise/counterclockwise asymmetry in conducting and mechanical properties of rotating (semi)conductors

M. N. Chernodub

Institut Denis Poisson UMR 7013, Université de Tours, 37200 France and Pacific Quantum Center, Far Eastern Federal University, Sukhanova 8, Vladivostok, 690950, Russia (Dated: April 11, 2021) We argue that certain (semi)conductors should exhibit asymmetry of their mechanical and conducting properties with respect to clockwise/counterclockwise rotation. We show that a cylinder made of a suitably chosen semiconductor coated in a metallic film and placed in the magnetic-field background can serve as a "rotational diode" which conducts electricity only at a specific range of angular frequencies. The critical angular frequency and the direction of rotation can be tuned with the magnetic field's strength. Mechanically, the rotational diode possesses different moments of inertia when rotated in clockwise and counterclockwise directions.

Introduction. Effects of gravity, rotation, and acceleration on electromagnetic and transport properties of physical systems have been a subject of intense interest throughout decades. [START_REF] Darling | The fall of charged particles under gravity: A study of experimental problems[END_REF] The Einstein-de Haas 2 and Barnett 3 effects relate mechanical torque and magnetization in ferromagnets. In metals, the uniform rotation acts on electrons via a centrifugal force that produces a small, but experimentally observable radial gradient of electric potential [START_REF] Beams | Potentials on Rotor Surfaces[END_REF] . The proposed rotational analogue of the classical Hall effect 5 highlights a well-known similarly of rotation with the magnetic field in non-relativistic systems. On the quantum level, rapidly rotating C 60 fullerenes are suggested to exhibit the Zeeman splitting in energy levels in the absence of a true magnetic field. [START_REF] Lima | Effects of rotation in the energy spectrum of C60[END_REF] Accelerating conductors generate intrinsic electric fields [START_REF] Moorhead | Electric fields in accelerating conductors: Measurement of the EMF in rotationally accelerating coils[END_REF] while gravity exerts a force on the electrons that induces the electric field outside a metal surface [START_REF] Witteborn | Experimental Comparison of the Gravitational Force on Freely Falling Electrons and Metallic Electrons[END_REF] . Gravitational forces are expected to lead to various thermoelectromagnetic effects in (super)conductors [START_REF] Anandan | Relativistic thermoelectromagnetic gravitational effects in normal conductors and superconductors[END_REF] . At the same time, the quantum Hall conductance, as a true topological quantity, turns out to be insensitive to background gravity [START_REF] Hehl | Is the Quantum Hall Effect Influenced by the Gravitational Field?[END_REF] . The inclusion of the spin degrees of freedom -and the ability to mechanically manipulate them in noninertial frames -is expected to play an important role in nano-electromechanical systems within the scope of the rapidly developing field of spintronics [START_REF] Matsuo | Mechanical generation of spin current[END_REF] . The emergence of synthetic gravitational fields in various condensed matter systems opens a new door for discoveries of novel quantum gravito-electromagnetic effects [START_REF] Vozmediano | Gauge fields in graphene[END_REF][START_REF] Volovik | The Universe in a Helium Droplet[END_REF] .

In our paper, we explore the mechanical and transport properties of rotating semiconductors and show that they break the equivalence of clockwise/counterclockwise rotations, which is naively expected for any isolated system. This purely quantum effect has its roots in a simple problem of classical electrodynamics.

Problem from classical electrodynamics. Let us consider an uncharged conducting cylinder of radius R and height L which rotates rigidly with the constant angular velocity Ω = Ω e z about its symmetry axis z. We place the cylinder in the background of constant and uniform magnetic field B = B e z directed along the axis of rotation as shown in Fig. 1(a). We set the relative permeability and permittivity of the material to unity ( = 1, µ = 1) and use Gaussian units.

Treating the system in the scope of classical electrodynamics, one can show that the interior of the cylinder accumulates a uniformly distributed electric charge with the bulk density: 14

ρ bulk = - ΩB 2πc , (1)
while the cylinder boundary (the part tangential to the axis z) acquires the uniform surface charge density:

ρ surf = RΩB 4πc . (2)
Since the net electric charge of the isolated cylinder is zero, the bulk (1) and surface (2) charges compensate each other exactly: πR 2 Lρ bulk + 2πRLρ surf = 0. We do not restrict the mutual directions of Ω and B so that the excess of the charge density (1) [and, respectively,

] can take both positive and negative values. The effect originates from to a finite conductivity σ = 0 of the rotating cylinder. In an equilibrium state of an isolated physical body, the Joule losses should be absent. This property immediately implies the absence of any dissipative electric currents in the system. In turn, the Ohmic dissipation is generated by a local electric current with respect to the ionic crystal lattice of the conductor. Therefore, the current should vanish in the corotating frame in which the conductor appears static. If the axis of magnetic field and the angular velocity vector align with each other, the local magnetic flux piercing the conductor is not affected by the mechanical constant with rotation. Consequently, the eddy (Foucault) currents and the associated energy losses are absent.

Denoting the quantities in the corotating (laboratory) frame by tilted (non-tilted) variables, the infinitesimal transformation between the coordinates in these frames reads as follows:

dr = dr -vdt, d t = dt, v = Ω × r , ( 3
)
where v is the local velocity of the fixed point r of the cylinder with respect to the laboratory frame. We con- sider a non-relativistic rotation which guarantees the validity of the causality constraint, |Ω|R c, where c is the speed of light.

The electromagnetic fields in the laboratory and corotating frames are related to each other as follows:

B = B, Ẽ = γ E + v c × B , (4)
where γ = 1/ 1 -v 2 /c 2 is the relativistic Lorentz factor. Hereafter, we ignore all relativistic corrections because they are negligibly small in this system. The rotational motion of the bulk (1) and surface (2) electric charges produces, via the Ampère law, an additional magnetic field which, however, will be dropped out in the following as it gives a tiny correction to the existing magnetic-field background B. Up to negligible relativistic effects, the densities (1) and ( 2) are the same in the laboratory and corotating frames, ρ = ρ. The absence of the current density in the corotating frame, J = σ Ẽ = 0, implies that the rotating conductor produces the radial electric field in the laboratory frame:

E = - v c × B = - ΩB c r ⊥ . (5)
In the last relation, we take into account the collinearity Ω B e z , and denote by r ⊥ the radial component of the coordinate, e z ⊥ r ⊥ , so that r = r ⊥ + ze z . The rotation-induced electric field (5) generates the uniform charge density in the interior of the cylinder, ρ = ∇ • E/(4π), providing us with the result (1). The requirement of the global charge neutrality leads, in turn, to the accumulation of the uniform surface charge (2) at the edge of the cylinder. The charge density is qualitatively shown in Fig. 1(b). In real metals, the width w of the surface layer is extremely small (w R) being of the order of a few nanometers.

Rotation and band filling. Our paper is based on the simple observation that a mechanical rotation in the background of the collinear magnetic field leads to a shift of the Fermi energy ε F due to the uniform, coordinate- independent accumulation of the electric charge density (1) in the bulk of the system. We analyze the situation that happens when the rotation drives the Fermi energy across the edge of the conductance or valence band. We demonstrate that this crossing breaks the discrete clockwise/counterclockwise rotational symmetry Ω → -Ω for cylinders made of a semiconducting material. The effect impacts the mechanical and conducting properties of the system.

Before proceeding further, we mention that the interplay between the rotation and the magnetism discussed in this paper has no direct relation to the Einstein-de Haas effect 2 (which demonstrates the appearance of a mechanical torque exerted by an external magnetic field on a ferromagnet) and the Barnett effect 3 (which reveals a reciprocal phenomenon: a mechanical rotation changes the magnetization of a spinning ferromagnet). These phenomena appear naturally as a consequence of the conservation of angular momentum. They demonstrate a close relationship between the magnetism, induced by the spin and the orbital motion of the electrons, and the mechanical rotation. However, both the Einstein-de Haas and Barnett effects are odd under the time reversal transformation which keeps the rotational symmetry Ω → -Ω unbroken (see, for example, the experimental work [17]). On the contrary, we consider the effect that breaks the clockwise/counterclockwise symmetry of the rotation.

Given the generic nature of the effect, it is sufficient to consider a degenerate semiconductor with a simple parabolic form for conduction and valence energy bands, respectively:

16 ε (e) k = ε G + k 2 2m e , ε (h)
k = - k 2 2m h . (6)
Here m e (m h ) is an effective mass of electrons (holes) and ε G is the gap between valence and conduction bands. We neglect Zeeman and spin-orbit interactions which do not play a significant role in the effect. We consider a device made of an intrinsic (undoped) p-type semiconductor with a fully filled valence band. The Fermi energy lies in the gap close to the edge of the valence band, as shown in Fig. 2(a). In conventions of Eq. ( 6), the zero Fermi energy, ε F = 0, corresponds to the upper edge of the valence band. We assume that the temperature is sufficiently low so that thermal energy is smaller than the energy gap between the bands, k B T ε G . We also coat the cylindrical semiconductor with a thin cylindrical shell (Fig. 3) made of a metal with a wide enough conduction band that includes the Fermi energy level ε F = 0. Therefore, the interior charge accumulation (1) happens inside the semiconducting bulk while the boundary charge buildup (2) occurs within the thin metallic layer. The electrodes are connected only to the interior (semiconducting) part of the device and do not touch the metallic coating.

We apply the background magnetic field along the z axis, B = Be z (with B > 0) and assume that the magnetic field is sufficiently weak so that the semiconductor band spectrum (6) serves as a good approximation to the problem.

Conductivity and rotation. With the fully filled valence band, empty conduction band, and a wide energy gap between these two bands, the interior of the static (non-rotating) cylinder resides in an electrically insulating state, Fig. 2(a). The electrons cannot be thermally excited from valence band to the conduction band.

The clockwise rotation (Ω < 0) makes the bulk charge density (1) positive implying that a part of electrons is relocated from the interior of the cylinder to its metallic boundary (2). The external metallic coating serves as a reservoir which accommodates the electrons displaced from the interior of the system. The rotation lowers the Fermi level in the bulk thus creating unfilled states near the Fermi level. The system enters the conducting regime, Fig. 2(b).

The shift of the Fermi energy due to rotation,

ε F (Ω) = - 2 2m h 3πΩB 2c 2/3 0, ΩB 0 , (7)
is determined by a comparison of the bulk density (1) with the density of the degenerate gas of fermions (in our case, holes):

ρ = k 3 F 3π 2 3 , k F = √ -2m h ε F . (8)
Thus, the clockwise rotation (Ω < 0) turns the interior of the cylinder from the insulator into a conductor phase. For a slow rotation with

|ε F (Ω)| k B T ε G , (9)
the thermally excited electrons from the Fermi sea will fill in the hole pocket in the valence band. Each liberated energy creates one electron and one hole charge carriers with the equal number density n = p = ρ bulk /(-e). The conductivity of the system becomes as follows:

σ = - (µ e + µ h )ΩB 2πc > 0, ΩB < 0 ( 10
)
where µ e (µ h ) is the electron (hole) mobility. We used the expression (1) for the carrier concentration induced by the combined effect of magnetic field and rotation in the semiconducting bulk. Notice that the effect (10) has a universal character in a sense that it does not depend on the details of the band structure (6) provided the rotation does not shift the Fermi energy ( 7) across the boundary of the band and the hierarchy (9) holds. The counterclockwise rotation (Ω > 0) induces the radial electric field which tends to make the bulk charge density (1) negative via displacing the electrons from the exterior metallic coating to the semiconductor interior. However, despite the metallic coating of the cylinder has enough supply of electrons, the semiconducting interior cannot accommodate them because the valence band is already fully filled, Fig. 2(c). Therefore, the interior remains in the insulating state 15 :

σ = 0, ΩB 0. ( 11
)
The clockwise/counterclockwise asymmetry of the device is a purely quantum phenomenon based on the Pauli exclusion principle. Using symmetry arguments one show that the effect has its roots in the absence of a definite parity with respect to the time-reversal transformation, T : t → -t. Indeed, the sign flip of the (T-odd) angular velocity Ω does not bring the device to the same state because the magnetic field B has a T-odd parity while the electric charge density ρ, accumulated due to the collective effect of magnetic field and rotation, is a T-even quantity:

T : B → -B, Ω → -Ω, ρ → ρ . ( 12
)
The dependence of the conductivity on the angular frequency, Eqs. ( 10) and (11), is shown in Fig. 4: the cylinder made of a semiconductor material with the threshold chemical potential behaves as an insulator for the rotation in the counterclockwise sense (Ω > 0) and a conductor when it turns in the counterclockwise direction (Ω > 0). The directions are inverted with the flip of the sign of the magnetic field. 13). The linear slopes are determined by the background magnetic field B > 0, Eq. ( 10).

A similar effect appears in intrinsic n-type semiconductors where the Fermi energy lies near the edge of the conduction band, ε F = ε G . Its phase diagram is reverted with respect to the direction of rotation, Ω → -Ω.

Thus, the idea behind our mechanism is simple: a background magnetic field tends to shift the Fermi energy (the chemical potential) in the bulk of the rotating system thus affecting the conductivity of the latter. It can also be applied to semimetals metals with the Fermi energy lying in the vicinity of the upper edge of the hole pocket (the valence band) or above but close to the lower edge of the conduction band. The rotation shifts the Fermi energy across the edge of the corresponding band and thus alters its conductivity. Denoting the gap ∆ F = -F > 0 and ∆ F = F -G > 0 in the former and latter cases, respectively, we obtain that the conductorinsulator transition takes place at the nonzero critical angular frequency with the magnitude:

Ω c = 2(2m∆ F ) 3/2 3π 3 B , ( 13
)
where m is the mass of the appropriate carrier. The (hole) conductivity is shown in Fig. 4 by the dashed line.

Mechanical properties. The effect may also have interesting mechanical consequences. The reorganization of the electric charge density in rotating conductors produces a supplementary angular momentum δL = L mech + L e.m. which adds up to a purely mechanical quantity L 0 associated with the rotation of the ionic lattice together with the original electrons. There are two contributions: the mechanical part coming from the mass redistribution of the displaced electrons L mech and a part originating from the angular momentum stored in the electromagnetic fields L e.m. .

In typical metals, the surface charge density is concentrated within a thin surface skin of the depth of a few nanometers. For practical mechanical calculations in macroscopic, centimeter-sized systems, the surface charge (2) can be treated as a δ-functional distribution at the edge of the system, r ⊥ = R. The same applies also to semiconductors, where the screening length lies in the micrometer range. Thus the electric charge density in the cylinder may be approximated by the following function:

ρ e (r) = ΩB 4πc [Rδ(r ⊥ -R) -2] , (14)
where we took the convention R 0 δ(r ⊥ -R)r ⊥ dr ⊥ = R. The mechanical excess of the angular momentum,

L mech = V d 3 r ρ m (r) r × v(r) = m e LΩ 2 R 4 4ce B, (15)
is determined via the surplus of the mass density

ρ m (r) = m e e ρ e (r) , (16)
where e = +|e| is the elementary electric charge and m e is the mass of an electric charge carrier. Here, for simplicity, we assume the presence of a single carrier and we set its mass to the electron mass. The mechanical angular momentum (15) originates from the displacement of the electrons from the bulk to the boundary (or viceversa, depending on the mutual orientation of the angular momentum Ω and the background magnetic field B).

Notice that since the displaced mass is proportional to the angular frequency, Eq. ( 14) and ( 16), the kinetic momentum ( 15) is an even function of Ω.

The local angular momentum carried by the electromagnetic field L e.m. (r) = r × S can be expressed via the Poynting vector S = E × B. This calculation, however, poses a practical inconvenience as it requires the integration of the local momentum over the whole spatial volume and involves the calculation of the electric field E = -∇φ [typically done via the electrostatic potential φ = φ(r)] in the exterior of the cylinder. An equivalent definition of the electromagnetic angular momentum for spatially finite systems is based on the Maxwell form:

L e.m. = 1 c d 3 r ρ e (r) r × A = LB 2 R 4 8c 2 Ω , (17)
which involves the vector potential A in the Coulomb gauge, ∇ • A = 0. In the evaluation of Eq. ( 17), we set B = ∇ × A with A = (Br ⊥ /2)e ϕ where e ϕ = e z × e ⊥ . The energy stored in the induced electrostatic field inside the cylinder (5), calculated in the laboratory frame, E e.m. ≡ E (E) e.m. =

1 8π V d 3 r E 2 (r) = Ω 2 B 2 R 4 16c 2 , ( 18
)
is related to the angular momentum (17) • dΩ. Finally, the energy stored by the magnetic field is insensitive to rotation (up to a tiny correction due to an extra magnetic field generated via the Ampère circular current).

As we discuss later, the kinetic angular momentum (15) associated with the displaced mass (16) in nonrelativistic systems is much smaller than the angular momentum stored in the electromagnetic fields (17). Therefore, we ignore the mechanical angular momentum in our discussion below and take for the excess of the angular momentum its electromagnetic part only, δL = L e.m. . The electromagnetic moment of inertia I e.m. can be defined either via the angular momentum, L e.m. = I e.m. Ω, or, equivalently, via the energy: I e.m. = ∂ 2 E e.m. /∂Ω 2 :

I e.m. = 1 8π V d 3 r B 2 (r) = LR 4 8c 2 B 2 . ( 19
)
In ordinary conductors, the extra angular momentum ( 17) is an odd function of the angular frequency Ω: the angular momentum changes its sign, L e.m. → -L e.m. , under the flip of the direction of rotation, Ω → -Ω. The reason behind this symmetry is obvious: the rotations in opposite directions lead to the appearance of the radial electric fields of equal magnitudes (but of opposite signs) due to the displacement of electrons either from the bulk to the boundary or vice versa. The electric field emerges due to the depletion (or surplus) of the uniform electric charge density inside the bulk, depending on the direction of rotation.

At the threshold chemical potential µ = 0, the rotational diode generates the electric field for the particular rotational direction for which Ω • B < 0. The rotation in opposite direction (Ω • B > 0) does not produce the electric field in the bulk. Therefore, the rotational energy and angular momentum stored in the induced electric field differs for the rotations in the CW and CCW senses. For the device at the threshold value of the Fermi level, the difference between the angular momenta for the CW/CCW rotations is given by Eq. ( 19):

Interestingly, this quantity has a universal character in a sense that it depends only on the geometry of the device and the background magnetic field. While the odd nature of the effects and the simplicity of the device that hosts them may seem attractive, quantitative estimates, given below, challenge the suitability of these effects for an experimental detection since their magnitude is not exceptionally large.

Electric charge in the bulk. In the background of the moderate magnetic field B = 1 T , the cylinder rotating with the angular frequency 18 Ω = 100 s -1 accumulates in its interior the electric charge with the bulk density ρ bulk = -8.8 × 10 3 e/cm 3 . This is a small, but still nonnegligible number.

Conductivity. The rotation-induced conductivity (10) depends substantially on the mobility µ of the charge carriers. For a typical semiconductor with mobility µ e ∼ µ e ∼ 10 3 cm 2 /(V•s), an Ω = 100 s -1 rotation in the magnetic field background B = 1 T induces conductivity σ 10 -13 Ω -1 cm -1 which makes the interior of the device as "good" conductor as, for example, glass or rubber. This exceptionally bad conductivity can, however, be improved in systems with a high mobility of charge carriers (possibly gated to achieve the correct position of the Fermi level at the upper/lower edge of a hole/electron band so that the non-rotating system resides at the border of an insulating state). For example, for AlGaAs/GaAs heterostructures featuring the high-mobility two-dimensional electron gas with 19 µ = 3.5 × 10 7 cm 2 /(V•s), the rotation induces conductivity compared to the lowest value σ 10 -8 Ω -1 cm -1 achievable in a pure semi-insulating GaAs crystal [START_REF] Mccluskey | Dopants and Defects in Semiconductors[END_REF] .

Angular momentum. The kinetic angular momentum associated with the radial displacement of electrons ( 15) is much smaller than the angular momentum stored in the electromagnetic field (17): L mech /L e.m. ∼ 10 -9 . Therefore, the mechanical properties of the device are determined only by the electromagnetic fields generated by the rotation. The difference between the angular moments of inertia for clockwise and counterclockwise rotation (20) of a centimeter-sized cylinder (L = R = 1 cm) in the magnetic field B = 1 T is ∆I e.m. 10 -15 g • cm [START_REF] Einstein | Experimental proof of the existence of Ampere's molecular currents[END_REF] . No matter how small this number may seem from the first sight, it corresponds to a moment of inertia of a water droplet in a typical fine fog (with the size about 10µm) which is already a macroscopic objet. The latter example gives us some hope that the change in the CW/CCW moments of inertia may be within experimental reach despite it constitutes a negligible fraction of the total moment of inertia of the system, ∆I e.m. /I tot e.m. 10 -16 .

Summary. We demonstrated that the absence of a definite time-reversal state in a mechanically rotating semiconductor in the background magnetic field leads to asymmetry of its mechanical and transport properties with respect to rotations in clockwise and counterclockwise directions. A fine-tuned system becomes a "rotational diode" that possesses different moments of inertia and resides in different conductor/insulator phases when rotated in opposite directions. Although the effect bears its roots in classical electrodynamics of rotating conductors in the background magnetic field, the CW/CCW rotational asymmetry appears as a purely quantum phenomenon based on the Pauli exclusion principle. The estimated magnitude of these effects turns out to be rather small for realistic materials.

FIG. 1 .

 1 FIG. 1. (a) Rotating conducting (metallic) cylinder in the magnetic field background. (b) Qualitative behavior of the electric charge density inside the cylinder (with w R).

FIG. 2 .

 2 FIG. 2. The effect of rotation on the band filling of a semiconductor rotating with the angular velocity Ω = Ωez in the background of magnetic field B = Bez with B > 0.

FIG. 3 .

 3 FIG. 3. The rotational diode.

FIG. 4 .

 4 FIG.4. The conductivity σ of the rotational diode vs. angular frequency Ω. The solid line corresponds to a p-type semiconductor at the threshold Fermi level, εF = 0. The dashed line gives the generic case (0 < εF < εG) with a nonzero critical angular frequency Ωc, Eq. (13). The linear slopes are determined by the background magnetic field B > 0, Eq. (10).

  via the thermodynamic relation dE e.m. = Ω • dL e.m. , as expected. The energy in the corotating frame, Ẽe.m. = E e.m. -Ω • L e.m. , satisfies the relation, d Ẽe.m. = -L e.m.

  ∆I e.m. ≡ I CW e.m. -I CCW e.m. = -LR 4 8c 2 B 2 sign (Ω • B) .

ACKNOWLEDGMENTS

The author is grateful to Karl Landsteiner and María Vozmediano for correspondence and interesting discussions. The work is supported by Grant No. 0657-2020-0015 of the Ministry of Science and Higher Education of Russia.