Qing Xiang Xiong
email: qingxiang.xiong@insa-rennes.fr

Fekri Meftah

Determination on pore size distribution by a probabilistic porous network subjected to salt precipitation and dissolution

Keywords: pore size distribution, waterfront, probabilistic method, transformation laws, salt precipitation, salt dissolution

Pore system strongly affects durability of building materials. Therefore, representative models for pore system and evolution of the pore system during aging are important for understanding physical processes that affect durability of building materials. In this study, a porous network model was developed based on a probabilistic method to describe pore size distribution, and three transformation laws were employed to establish relationship between the initial pore sizes and modified pore sizes (after salt precipitation and dissolution) regarding the waterfront. Proposed model illustrated evolution of pore size distribution due to salt precipitation and dissolution based on overall porosity and water saturation degree. Moreover, the model can be used to implement the filling/emptying ratio of pores at vicinity of waterfront. Successive transformations were introduced into the model to illustrate evolution of pore size distribution due to long-term salt intrusion. Lastly, parametric studies on the model were performed to further elucidate the modifications on pore size distribution due to salt precipitation and dissolution.

Introduction

Pore structure of a construction material is an intrinsic property which is closely related to durability and service life of building structures. Salt precipitation and dissolution are notorious deteriorating factors of construction materials, particularly the materials with highly interconnected pores which facilitate transport of moisture and aggressive chemical species [START_REF] Huang | Numerical simulation of moisture transport in concrete based on a pore size distribution model[END_REF][START_REF] Ahl | Salt diffusion in brick structures Part II The effect of temperature, concentration and salt[END_REF]. Salt precipitation and dissolution critically alter microstructure of porous materials [START_REF] Gentilini | Effect of salt crystallisation on the shear behaviour of masonry walls: An experimental study[END_REF][START_REF] Veran-Tissoires | Discrete Salt Crystallization at the Surface of a Porous Medium[END_REF]. Nucleation and propagation of salt in pore space during salt precipitation generate pressure on the pore wall, which produces fractures on or through the pores [START_REF] Feraille-Fresnet | Thermo-hydro-chemical modelling of a porous medium submitted to high temperature: An application to an axisymmetrical structure[END_REF]. Subsequent salt dissolution increases porosity and accelerates penetration of fluid through the pores, which ultimately reduces durability of construction materials [START_REF] Fenaux | Modelling the transport of chloride and other ions in cement-based materials[END_REF][START_REF] Kumar | Porosity, pore size distribution and in situ strength of concrete[END_REF]. As suggested by George et al. [START_REF] Scherer | Crystallization in pores[END_REF] and Coussy et al. [START_REF] Coussy | Deformation and stress from in-pore drying-induced crystallization of salt[END_REF], salt precipitation and dissolution are generally unavoidable. Thus, understanding the mechanisms by which salt deteriorates construction materials is the key to ascertain durability of these materials. Evaluating modifications on pore structure due to salt precipitation and dissolution is the first step towards attaining such understanding.

Porosity is an overall measure of pore structure, which gives ratio of pore volume to total volume of a porous material. However, porosity does not give any information about pore structure: pore size, shape or distribution [START_REF] Day | Measurement of porosity in blended cement pastes[END_REF]. Two porous materials with similar porosity may have entirely different pore structures [START_REF] Dimri | Fractal Models in Exploration Geophysics: Applications to Hydrocarbon Reservoirs[END_REF]. Thus, pore size distribution needs to be considered when investing phenomena that modify the pore structure [START_REF] Benavente | Role of pore structure in salt crystallisation in unsaturated porous stone[END_REF].

Salt precipitation and dissolution are strongly affected by water content of porous material [START_REF] Castellazzi | A coupled multiphase model for hygrothermal analysis of masonry structures and prediction of stress induced by salt crystallization[END_REF]. At macroscopic scale, precipitated salt can be observed at the vicinity of drying front (i.e., boundary between saturated and non-saturated zones) inside porous material when specimen subjected to subjected to up-taking and drying cycles of saline solution [START_REF] Scherer | Stress from crystallization of salt[END_REF]. Even if no explicit experimental evidence exists to the authors' knowledge, the same schema can be adopted at pore scale. Indeed, a given degree of water saturation corresponds to the location of waterfront at a pore size that coincides with the radius of water capillary meniscus [START_REF] Huang | Numerical simulation of moisture transport in concrete based on a pore size distribution model[END_REF]. Therefore, one may extrapolate that salt precipitation and dissolution occur at the vicinity of waterfront within pore structure [START_REF] Benavente | Role of pore structure in salt crystallisation in unsaturated porous stone[END_REF]. On the basis of this inference, precipitation and dissolution mainly affect pores at close proximity to waterfront. Thus, modifications of pore size may predominantly restrict to a certain pore population with size close to the radius of capillary meniscus. In this case, overall porosity is not sufficient to describe elaborated modifications occurring near the waterfront at pore scale. Thus, detailed evolution of pore size distribution needs to be considered together with overall porosity and water saturation degree.

From experimental point of view, there are several experimental methods dedicated to characterize both overall porosity and pore size distribution [START_REF] Diamond | A critical comparison of mercury porosimetry and capillary condensation pore size distributions of portland cement pastes[END_REF][START_REF] Roels | Characterisation of pore structure by combining mercury porosimetry and micrography[END_REF][START_REF] Gallucci | 3D Experimental Investigation of the Microstructure of Cement Paste Using Synchrotron X-Ray Microtomography (mu CT)[END_REF]. However, sample pre-treatments such as sample drying are generally required prior to analysis of pore size distribution when adopting experimental methods [START_REF] Aligizaki | Pore Structure of Cement-Based Materials: Testing, Interpretation and Requirements[END_REF], which may alter water profile of pore structure. Moreover, some experimental methods are destructive that may damage pore structure of the investigated material [START_REF] Shi | Contact angle and damage during mercury intrusion into cement paste[END_REF]. Thus, experimental methods face the dilemma that how to characterize pore size distribution and at the same time allow water-dependent evolution of pore size distribution within porous material. Indeed, no convincing and experimentally verified answer yet exists on quantitative measurement of pore size distribution in the course of salt precipitation and dissolution. For instance, some consider only the behavior of crystallization relevant to solution properties [START_REF] Rodriguez-Navarro | Salt weathering: influence of evaporation rate, supersaturation and crystallization pattern[END_REF], while others still focus on the study of weight loss and degradation of visual appearance due to salt precipitation [START_REF] Benavente | Quantification of salt weathering in porous stones using an experimental continuous partial immersion method[END_REF]. Such studies may generate results that cannot readily be compared with our lack of knowledge concerning quantitative measurement of pore size distribution in the course of salt precipitation and dissolution. Moreover, experimental methods are indeed out of the scope of this paper. This paper is aiming at proposes a modeling approach in order to describe elaborated evolution of pore size distribution of porous material when subjected to salt precipitation and dissolution. Pore size distribution is a complex property that encompasses pore size with several orders of magnitude. Fig. 1 illustrates that there are three classes of pores in cement-based material based on the pore radius: gel pores (0.25 nm -5 nm ), capillary pores (2.5 nm -5 μm), and air voids (> 5 µm) [START_REF] Gong | Empirical Estimation of Pore Size Distribution in Cement, Mortar, and Concrete[END_REF]. Gel pores are generally detected by nitrogen or water adsorption method. Capillary pores and air voids are evaluated by mercury intrusion porosimetry and image analysis, respectively. Fig. 1. Classification of pore sizes in cement-based material. [START_REF] Scherer | Stress from crystallization of salt[END_REF] In spite of heterogeneities in pore size, pore size distribution obeys a statistical behavior [START_REF] Brutsaert | PROBABILITY LAWS FOR PORE-SIZE DISTRIBUTIONS[END_REF], which indicates that shape of pore size vs. pore size-frequency curve follows a probabilistic model such as Raleigh-Ritz (R-R) distribution [START_REF] Maekawa | Multi-scale Modeling of Concrete Performance[END_REF], Weibull distribution [START_REF] Lu | Fracture statistics of brittle materials: Weibull or normal distribution[END_REF], normal distribution [START_REF] Lu | Fracture statistics of brittle materials: Weibull or normal distribution[END_REF], or lognormal distribution [START_REF] Shi | Lognormal Simulation of Pore Size Distributions in Cementitious Materials[END_REF]. Among these models, lognormal distribution was reported to yield the most accurate estimation on pore size distribution of cementitious materials [START_REF] Hou | Statistical modelling of compressive strength controlled by porosity and pore size distribution for cementitious materials[END_REF][START_REF] Chen | Influence of water-to-cement ratio and curing period on pore structure of cement mortar[END_REF].

The earliest work on fitting lognormal distribution into experimental pore size distribution was presented by Hatch et al. [START_REF] Hatch | Statistical description of the size properties of non uniform particulate substances[END_REF] and Austin [START_REF] Austin | Methods of representing distribution of particle size[END_REF]. In 1971, Diamond et al. [START_REF] Diamond | Generalized log-normal distribution of pore sizes in hydrated cement paste[END_REF] developed a generalized lognormal distribution to characterize pore size distribution of cement mortars. In 1991, Shi et al. [START_REF] Shi | Lognormal Simulation of Pore Size Distributions in Cementitious Materials[END_REF] proposed a multimodal lognormal distribution model to simulate pore size distribution of cement mortars. In 1992, Holly et al. [START_REF] Holly | Modelling relationships between permeability and cement paste pore microstructures[END_REF] established a porous network model based on multi-lognormal distribution to describe effect of alterations in microstructure on permeability of cement mortars. In 2013, Chen et al. [START_REF] Chen | Influence of water-to-cement ratio and curing period on pore structure of cement mortar[END_REF] demonstrated that compound lognormal distribution was the most effective among selected statistical models to characterize porous structure. Recently, Jiang et al. [START_REF] Jiang | Modelling of water vapour sorption hysteresis of cement-based materials based on pore size distribution[END_REF] adopted a triple-compounded lognormal distribution to predict hysteresis of water vapour sorption in cement-based materials. These literature reports support that lognormal distribution, which is applicable exclusively for positive pore radius, is powerful to model pore size distribution of cement-based materials [START_REF] Zydney | Use of the log-normal probability density function to analyze membrane pore size distributions: functional forms and discrepancies[END_REF]. Therefore, in this study, we adopted lognormal distribution to describe initial pore size distribution prior to salt precipitation and dissolution.

Based on a premise that initial pore size distribution follows lognormal distribution, this research aims to develop a probabilistic method to describe modified pore size distribution concerning waterfront induced by salt precipitation and dissolution. The probabilistic method is achieved by accomplishing the transformation models between pore radius before and after salt precipitation and dissolution. On the basis of the proposed method, modifications of pore size distribution induced by the alterations of water saturation degree and porosity are investigated. Most importantly, the proposed method is able to describe the successive transformations of pore size distribution and filling/emptying ratio due to the long-term salt precipitation and dissolution. These merits are rarely reported in literature. This paper is organized as follows. Section 2 introduces probabilistic approach for pore size distribution and three transformation models for pore radius. Initial pore size distribution data derived MIP analysis and subsequent simulation of the experimental data by lognormal distribution are given in Section 3. Section 4 provides examples for application of the probabilistic method and transformation models. Comparisons among transformation models are available in this section. Some conclusions and discussions end the paper in the last section.

Probabilistic porous network

Porous material is regarded as a heterogeneous, multi-phase material (Fig. 2). Solid phase is composed of solid matrix and precipitated salt. Liquid phase is constituted of saline solution with water as the solvent and ionic species as the solute. Meanwhile, dry air and water vapour make up the gaseous phase. In most porous materials, especially cement-based materials, gel pores filled with water constitute a small proportion of pore population [START_REF] Holly | Modelling relationships between permeability and cement paste pore microstructures[END_REF]. Large pores exist as isolated cavities, and contribute little to fluid transport and chemical reactions [START_REF] Castellazzi | A coupled multiphase model for hygrothermal analysis of masonry structures and prediction of stress induced by salt crystallization[END_REF][START_REF] Zhang | Transport properties in unsaturated cement-based materials -A review[END_REF]. While interconnected capillary pores partially filled with water determine permeability and transport property of porous materials [START_REF] Holly | Modelling relationships between permeability and cement paste pore microstructures[END_REF]. Capillary interface (i.e., waterfront) which is often located at capillary pores, separates the liquid from gaseous phase. As suggested by Yokoyama et al. [START_REF] Yokoyama | The effects of pore size distribution and working techniques on the absorption and water content of concrete floor slab surfaces[END_REF], quantity of pores exceeding 100 nm in diameter strongly affects the water absorbency, while number of pores smaller than10 nm in diameter governs moisture content of a porous material, which verifies our hypothesis that liquid-gas interface is located at capillary pores. The capillary pores dominate the water saturation degree in our proposed model. Salt precipitation and dissolution are one of the most important deteriorating factors of pore structure [START_REF] Gupta | Paradoxical drying of a fired-clay brick due to salt crystallization[END_REF]. Precipitation and dissolution are complex phenomena that involve physical and chemical processes [START_REF] Celia | Recent advances in pore scale models for multiphase flow in porous media[END_REF][START_REF] Dashtian | Pore-network model of evaporation-induced salt precipitation in porous media: The effect of correlations and heterogeneity[END_REF]. During exposure of porous materials to an aggressive (i.e., high salt concentration) environment (relative humidity or temperature), soluble salt continuously penetrates the materials [START_REF] Coussy | Deformation and stress from in-pore drying-induced crystallization of salt[END_REF]. Due to change in environmental condition, water evaporates, which increases concentration of salt near the waterfront beyond supersaturation threshold concentration for precipitation [START_REF] Benavente | Role of pore structure in salt crystallisation in unsaturated porous stone[END_REF][START_REF] Benavente | Quantification of salt weathering in porous stones using an experimental continuous partial immersion method[END_REF]. Consequently, salt precipitates out of saline solution, nucleates, and grows on surface of pore wall near waterfront [START_REF] Scherer | Crystallization in pores[END_REF][START_REF] Scherer | Stress from crystallization of salt[END_REF]. Further change in environmental condition induces condensation of water vapour inside pores which dissolves precipitated salt and decreases salt concentration [START_REF] Coussy | Deformation and stress from in-pore drying-induced crystallization of salt[END_REF]. Therefore, salt precipitation and dissolution are a reversible process which repeated occurrence influences integrity and durability of pore structure.

The primary aim of this study is to develop a model for the water-dependent evolution of pore size distribution in the course of salt precipitation and dissolution. We substantially focus on the development of pore structure itself, chemical reactions and environmental conditions are not involved in this research.

Probabilistic method for pore size distribution

As suggested by Shi et al. [START_REF] Shi | Lognormal Simulation of Pore Size Distributions in Cementitious Materials[END_REF], initial pore size distribution of porous material is presented as a lognormal probability density function (PDF):

( ) ( ) 2 0 0 0 2 0 ln 1 ( ) exp 2 2 r r r r P r r µ σ σ π   -   = -     (1)
Correspondingly, the cumulative distribution function (CDF) is given by: ( ) ( )

0 0 ln 1 1 2 2 2 r r r F r erf µ σ   - = +     (2)
where is initial pore radius, is location parameter and is shape parameter.

The PDF must satisfy normalization conditions of probability theory:

0 0 0 0 ( ) 0 ( ) 1 P r P u du ∞ ≥    =   ∫ (3)
The parameters of function ( ) are determined by MIP analysis. According to Washburn equation [START_REF] Washburn | Note on a Method of Determining the Distribution of Pore Sizes in a Porous Material[END_REF], larger pores are filled initially, followed by smaller pores at increasing pressures. Minimum pore radius is determined by maximum pressure imposed on porous materials: the higher is the imposed pressure, the smaller is the detected pore size, and the greater is the cumulated pore volume [START_REF] Schabowicz | Non-Destructive Testing of Materials in Civil Engineering[END_REF]. Function 1 -F( ) represents cumulative fraction of pore volume which is occupied by pores with pore radius greater than .

Initial pore volume fraction of pores with radius ranges from zero to is expressed with the following function:

0 0 0 0 0 0 ( ) ( ) r r P u du φ φ = ∫ (4)
where is initial porosity of a representative porous medium. The porosity is defined as a ratio of total pore volume to bulk volume :

0 P V V φ = (5)
In this study, the pores were assumed cylindrical and interconnected. Hence, initial volume of pore with radius is calculated by:

0 2 0 P v r L π = ( 6
)
where L is pore length, and L keeps constant because salt precipitation and dissolution affect pore radius rather than pore length.

Due to salt precipitation and dissolution, pore radius changes and we assumed that initial pore radius undergoes erosion on a distance of ( ), as shown in Fig.

3 [START_REF] Feraille-Fresnet | Thermo-hydro-chemical modelling of a porous medium submitted to high temperature: An application to an axisymmetrical structure[END_REF]. Accordingly, the relationship between initial pore radius and modified pore radius is represented by transformation function T:

1 0 0 0 ( ) ( ) T r f r r r ε → = = + (7)
where is modified pore radius after salt precipitation and dissolution, ( ) is the eroded size of pore, when salt precipitation occurs, ( ) < 0; when salt dissolution occurs, ( ) > 0. ( 0 ) is the transformation function, which will be illustrated in Subsection 2.2. Correspondingly, when salt precipitation and dissolution transform porosity from to , the pore with radius of belonging to initial probability density function becomes the pore with radius of belonging to modified probability density function (Fig. 4). Therefore, pore volume fraction of modified pores with radius ranges from zero to is expressed as:

1 1 1 1 1 0 ( ) ( ) r r P u du φ φ = ∫ (9)
Meanwhile, modified volume of pore with radius is defined by:

1 2 1 P v r L π = (10)
A probabilistic method can be derived by assuming that ratio of porosity before and after salt precipitation and dissolution is equal to ratio of volume for pores with radius and :

1 1 1 0 0 0 d r dr d r dr φ φ = (11)
Modified probability density function after salt precipitation and dissolution is derived by substituting Eq. 4 and Eq. 9 into Eq. 11:

0 0 1 1 0 0 1 0 ( ) ( ) ( ) f r P r P r r φ φ = (12)
Substituting Eq. 12 to Eq. 9 gives:

1 0 0 0 ( ) ( ) ( ) f u df u P u du u du φ φ ∞   =     ∫ (13)
By adopting the probabilistic method, probability density function ( ) is derived after successive transformations of pore radius (n is the transformation number), of which the details on the successive transformations are provided in Appendix C.

Heretofore, we have developed a probabilistic method describing probability density functions for initial pore radius and modified pore radius after salt precipitation and dissolution. The next section defines linear and non-linear transformation models for Eq. 7.

Transformation models for pore radius

Linear transformation model. This model assumes that incremental function ( ) is proportional to initial pore radius, which indicates that salt precipitation and dissolution modify entire range of pore size. The incremental function ( ) is defined as:

0 0 0 ( ) 0 L r r r ε ε = ⋅ ≤ ≤ ∞ (14)
Accordingly, linear transformation for pore radius is expressed as:

1 0 0 0 ( ) L L T r f r r r ε → = = + (15)
where is a constant which denotes the magnitude of variation in pore radius, and it is obtained by substituting Eq. 15 into Eq. 13:

1 0 1 L φ ε φ = - (16)
Under linear transformation model, magnitude of variation in pore size is influenced particularly by porosity, at which pores are uniformly filled/emptied irrespective of location of waterfront.

Non-linear transformation models. Non-linear transformation models account for location of waterfront. In non-linear transformation models, saline solution partially fills pore space and forms waterfront, which separates saturated pores from the unsaturated pore counterparts. Assuming that the waterfront is located in pores with radius , opening zone towards the waterfront is denoted as β. The incremental function ( ) is given by: 0

0 ( ) ( ) r S r ε ε = ⋅ (17)
where ( ) is a sigmoidal function (0 ≤ ( ) ≤ 1), which can be defined by two transformation models: monotonic and non-monotonic transformation models.

Monotonic transformation model (MT). Fig. 5a shows that in monotonic transformation model, function ( ) assumes an S-shape. Modification on pore size distribution affects pore which radius is larger than -. More precisely, modification gradually increases in pores which radius is between and + . At pores with radius greater than + , the modification is regarded as homogeneous increment constant

. Thus monotonic transformation model is expressed as:

1 0 0 0 ( ) ( ) MT MT MT T r f r r S r ε → = = + (18)
Details on function ( ) and constant are provided in Appendix A and B, respectively. Details on transformation function " and parameter " are available in Appendix A and B, respectively.

Non-monotonic transformation model (NMT)

Water saturation degree

Water condensation occurs in pores with radius smaller than , therefore, water saturation degree where ' represents probabilistic density function before or after salt precipitation and dissolution. Eq. 20 shows a direct relationship between water saturation degree and waterfront. Based on Fig. 6, it is apparent that water saturation degree increases as waterfront moves to larger pores.

Experimental data for initial pore size distribution

Initial pore size distribution incorporated in models described in previous section is derived from experimental data of cumulative volume fraction and incremental pore volume distribution of masonry cement measured by MIP analysis. (Table 1) are obtained for deriving probability density function of initial pore distribution (Fig. 8).

Parametric investigation

Fig. 8 shows that the size of pores in masonry cement ranges from 0.0015 μm to 160.2 μm. As previously mentioned, the size of capillary pores ranges from 0.0025 μm to 5 μm (Fig. 1), which allows waterfront moves within the pores with radius in the range of [0.0025μm, 5 μm]. Correspondingly, the opening zone β should be satisfied within -≥ )' and + ≤ )*+ . For simplifying proposed model, a constant value of = 0.1μm was selected. In addition, it should be emphasized that the presented transformation models cannot yield negative pore radius value. Negative pore radius value indicates that pores are completely filled by excess amount of precipitated salt which may eventually induce fractures within porous material [START_REF] Koniorczyk | Numerical Modeling of Salt Transport and Precipitation in Non-Isothermal Partially Saturated Porous Media Considering Kinetics of Salt Phase Changes[END_REF][START_REF] Koniorczyk | Experimental and numerical investigation of sodium sulphate crystallization in porous materials[END_REF][START_REF] Benavente | Thermodynamic modelling of changes induced by salt pressure crystallisation in porous media of stone[END_REF].

Linear transformation

During salt precipitation and dissolution, pore size undergoes series of transformation in porosity that → → ⋯ → ' → .

Fig. 9a-b show linear transformation from initial pore size distribution. Salt precipitation induces a reduction in porosity from 0.20 to 0.05, which shifts initial probability density and cumulative volume fraction curve towards lower range of pore radius, and increases the probability density. This indicates that salt precipitation generates smaller pores and reduces larger pores. In contrast, an increase in porosity from 0.2 to 0.9 induced by salt dissolution shifts initial probability density and cumulative volume fraction curve towards higher range of pore radius, and increases the width of probability density curve, hence promoting formation of larger pores. In addition, probability density decreases with the increase in porosity. Linear transformation model considers modifications on every pore size, of which a higher alteration from initial porosity produces a more significant change in pore radius.

Although linear transformation model provides simplified understanding and good representation of salt precipitation and dissolution, this model is not an accurate representation of actual condition because it does not consider the dependence of salt precipitation and dissolution on waterfront.

Non-linear transformation

In contrast to linear transformation model which simultaneously affects all pore size, non-linear transformation models aim to represent water-dependent transformation of pore size. The non-linear transformation models involve single transformation and successive transformations of pore size distribution.

Single transformation of pore size distribution

We considered a generalized situation at which drying of a porous material shifts waterfront towards smaller pores. Salt crystals are stabilized in pore space close to waterfront and modify pores around the waterfront. On the other hand, when a porous material is wetted, waterfront moves towards larger pores and the precipitated salt near the waterfront is re-dissolved in saline solution. The water-dependent transformation models predict evolution of pore size distribution due to salt precipitation and dissolution. Fig. 10 and 11 illustrate monotonic and non-monotonic transformation of pore size distribution due to single occurrence of salt precipitation and dissolution. By substituting Eq. 12 into Eq. 20, the location of waterfront is deduced at given porosity and water saturation degree. Salt precipitation promotes decreasing porosity, which shifts waterfront towards lower range of pore radius. Simultaneously, probability density increases and number of small pores increases for both monotonic and non-monotonic transformation. In contrast, due to salt dissolution, waterfront moves towards larger pores as a consequence of increasing porosity and water saturation degree. Probability density decreases and amount of larger pores increases for both monotonic and non-monotonic transformation. The modifications on porosity also alter pore volume fraction (determined by Eq. 9), as illustrated in Fig. 12, the area denotes the integration of pore volume fraction. The more significant is the deviation from initial porosity, the larger is the filled/emptied area being created, particularly under non-monotonic transformation. Fig. 13 suggests that non-monotonic transformation generates higher filling/emptying ratio than monotonic transformation at similar water saturation degree and porosity during salt precipitation and dissolution. Due to salt precipitation (Fig. 13a), non-monotonic transformation generates a filling ratio of 61.6% when porosity decreases from 0.20 to 0.15. Comparatively, monotonic transformation produces a filling ratio of 24.9%. Similarly, due to salt dissolution (Fig. 13b), emptying ratio implemented by non-monotonic transformation is higher than that resulted from monotonic transformation counterpart (45.6% vs. 23% at 0.25 porosity). Non-monotonic transformation model simulates pore filling/emptying accounting for highly-localized (i.e., at vicinity of waterfront) occurrence of salt precipitation and dissolution, which provides a more favorable representation of actual pore clog due to salt precipitation.

Non-linear transformation models indicate that the pores affected by the homogeneous effect make a substantial contribution. Non-linear transformation of pore size distribution considers localization of salt precipitation and dissolution at vicinity of waterfront, and indicates higher filling/emptying ratio than linear transformation counterpart which generates a uniform distribution of salt precipitation and dissolution. Due to salt precipitation and dissolution, non-monotonic transformation model modifies pore size distribution from unimodal to bimodal for large deviation in porosity because pore filling/emptying introduces a new peak in probability density function curve, which consequently alters pore population structure (i.e., proportion of small and large pores). Modifications on pore size distribution from unimodal to bimodal and multi-modal have been systematically introduced by Shi et al. [START_REF] Shi | Lognormal Simulation of Pore Size Distributions in Cementitious Materials[END_REF] and Holly et al. [START_REF] Holly | Modelling relationships between permeability and cement paste pore microstructures[END_REF]. In conclusion, non-linear transformation models more accurately represent the actual salt precipitation and dissolution which occur at close proximity to waterfront.

Successive transformations of pore size distribution

Under successive transformations, modifications on pore size distribution are generated by several occurrences of salt precipitation and dissolution (please refer to Appendix C for expressions of successive transformations). The modifications on pore size distribution are cumulative, of which existing pore size distribution is determined by preceding pore size distribution. Two examples were considered (Table 2): salt precipitation which decreases porosity from 0.20 to 0.15 after 30 successive transformations, and salt dissolution which increases porosity from 0.20 to 0.25 after 30 successive transformations.

a. Parameter 0 versus porosity

Constant denotes a magnitude of change in pore radius. Detailed definition on ε in successive transformations is available in Appendix C. When is less than zero, salt precipitation occurs. On the other hand, when is greater than zero, precipitated salt is re-dissolved. The higher is the value of | |, the more significant is the modification on pore radius.

During salt precipitation (Fig. 14a), decrease in porosity reduces value of . On the other hand, during salt dissolution (Fig. 14b), increase in porosity initially increases value of significantly. However, the value of progressively decreases before stabilizes. During salt precipitation at water saturation degree of 0.4, waterfront is located in small capillary pores, and there is small difference in value of for monotonic and non-monotonic transformation models. Increase in water saturation degree to 0.5 shifts waterfront to larger pores. Value of derived from non-monotonic transformation differs greatly from that derived from monotonic transformation. These results reveal that modification on pore radius is affected by water saturation degree. However, it should be pointed out that the values of calculated by proposed models are applicable to porosity which generates positive pore radius. Negative pore radius indicates completely filled pores, which produces localized damage near waterfront.

b. Evolution of pore size distribution

This section compares modifications of pore size distribution due to single transformation (where porosity increases/decreases in one-step) and 30 successive transformations (where porosity gradually increases/decreases by several steps) induced by salt precipitation and dissolution.

The comparisons between single transformation and successive transformations are shown in Fig. 15a-d. Under monotonic transformation, it is not surprising that there is no significant difference in modification on pore size distribution due to single transformation and successive transformations. Similarly, water saturation degree does not have noticeable influence on pore size distribution. Under non-monotonic transformation, single transformation generates bimodal distribution. On the other hand, successive transformations display a prominent increase in probability density near waterfront. Consequently, filling/emptying ratio generated from successive transformations is higher than that generated from single transformation for both monotonic and non-monotonic transformation, except for salt dissolution under non-monotonic transformation. Overall, these findings suggest that cumulative modifications on pore size due to successive salt precipitation should not be overlooked. Successive transformation model illustrates evolution of pore size distribution due to long-term exposure of porous material to saline solution. Salt precipitation and dissolution do not alter range of pore size, but change the proportion of small and large pores. Successive transformation model illustrates modifications on pore size distribution due to actual non-homogeneous salt precipitation and dissolution, which predominantly occur at close vicinity of waterfront and influenced by porosity and water saturation degree.

c. Evolution of water saturation degree due to successive transformations.

Under successive transformations, although waterfront is supposed to be in equilibrium, water saturation degree is affected by cumulative effects of successive transformations due to salt precipitation and dissolution. As shown in Fig. 16, due to salt precipitation (Fig. 16a), water saturation degree derived from successive monotonic transformation progressively increases, while water saturation degree derived from successive non-monotonic transformations decreases with decreasing porosity. On the other hand, due to salt dissolution (Fig. 16b), water saturation degree derived from successive monotonic transformation reduces slightly while successive non-monotonic transformation gradually increases water saturation degree with increasing porosity.

It is worth noting that this research assumes that liquid-vapor interface (i.e., waterfront) remains in equilibrium state in spite of the variations in porosity. Nevertheless, salt precipitation and dissolution are complex phenomena involving mass transfers and chemical reactions. Formation of salt crystals generally accompanies alterations in environmental conditions (salt concentration, pressure, relative humidity, and atmospheric temperature) [START_REF] Zhang | Multiscale lattice Boltzmann-finite element modelling of chloride diffusivity in cementitious materials. Part II: Simulation results and validation[END_REF][START_REF] Benavente | Salt influence on evaporation from porous building rocks[END_REF]. Changes in these environmental factors easily disturb the equilibrium state at waterfront [START_REF] Guimarães | Determination of chloride diffusivity through partially saturated Portland cement concrete by a simplified procedure[END_REF]. Hence, dynamic effects in environmental factors should be accounted for to consistently estimate water saturation degree at different porosity in future.

Conclusions

By studying water-dependent evolution of pore size distribution induced by salt precipitation (pore filling) and dissolution (pore emptying), a probabilistic method is established regarding the probability density functions for initial and modified pore-size distributions. This method is achieved by accomplishing the transformation models that determine a magnitude of change in pore radius, namely the parameter ε. One linear transformation model and two non-linear transformation models are constructed as a bridge connecting the initial pore radius and modified pore radius during salt precipitation and dissolution. Linear transformation model does not adequately characterize modifications on pore size distribution due to salt precipitation and dissolution. In comparison, non-linear transformation models (monotonic transformation model and non-monotonic transformation model) allow estimation on water-dependent modifications of pore size. Monotonic transformation model permits the modifications to occur in all unsaturated pores. While non-monotonic transformation model restricts the modifications to pores at close proximity to waterfront. Furthermore, non-monotonic transformation leads to bimodal pore size distribution and higher filling/emptying ratio than the monotonic transformation counterpart, suggesting that non-monotonic transformation model more favorably represents pore clog due to salt precipitation and dissolution.

Successive transformations of pore size distribution are developed to describe the evolution of pore size distribution due to long-term exposure of porous materials to saline solution. The results suggest

≤ ≤ -     - -  = = - + -< ≤ +          + < < ∞  (A.1)
The symmetric S-shape function "

for non-monotonic transformation law is given by:

( ) ( ) ( ) ( ) ( ) 0 3 0 0 0 0 3 0 3 0 0 0 0 3 0 0 0 2 2 2 2 3 1 4 3 2 2 2 2 2 3 1 4 3 2 0 c c c c c NMT c c c c c r r r r r r G r r r r S r r r r r K r r r r r r β β β β β β β β β β β β ≤ ≤ -     + - + -  = - + -< ≤        =    -- --  = - - + < ≤ +         + < < ∞  (A.2)

Appendix B. Calculation of parameter 0

For linear transformation, the modified pore size distribution is deduced as:

( )

0 1 1 0 0 1 ( ) 1 ( ) L P r P r φ ε φ = + (B.1)
Then the porosity fraction function yields:

( )

2 1 0 0 0 0 ( ) 1 ( ) L r P u du φ φ ε ∞ = + ∫ (B.2)
As to the fact that 2 (3)4u 6 = 1 , thusly, equation (B2) can be solved to determine the parameter for linear transformation model:

0 1 1 L φ ε φ = - (B.3)
For non-linear transformations, modified pore volume fraction function (B2) can be rewritten by definite integral:

( )

2 1 0 1 2 1 I I φ φ ε ε = + + (B.4)
Solving equation (B4) could obtain the expression of parameter :

2 1 1 1 2 0 2 4 1 2
I I I I φ φ ε   -± - -     = (B.5)
where 7 , 7 8 are definite integrals which are determined respectivily according to monotonic transformation law and non-monotonic transformation law. For monotonic transformation law, we have:

Appendix C. Pore size distribution for successive transformations

According to equation [START_REF] Benavente | Role of pore structure in salt crystallisation in unsaturated porous stone[END_REF], we can also deduce the pore size distribution after n+1 times of successive transformations by adopting successive monotonic transformation law and non-monotonic transformation law, respectively: For the monotonic transformation, the probability density function can be written as follows:

+ = + + + + = +    ≤ ≤ -  + ⋅  = -< ≤ +    +  + < < ∞   ∏ ∏ (C.1)
where is the pore radius after n times of transformation, defining as: The pore size distribution implement by the non-monotonic transformation is given by:

( ) ( ) ( ) ( ) 0 1 0 0 1 0 1 1 1 0 0 1 0 ( ) 0 ( ) ( ) ( ) ( ) ( )
+ + + = + ≤ ≤ -   + ⋅  -< ≤   =  + ⋅  < ≤ +    + < < ∞  ∏ ∏ (C.3)
The global porosity fraction function ( ) after successive transformations is defined as: It can be reorganized as: So that parameter can be calculated by the equation: in which 9 , 9 8 and 9 : are definite integrations, which are calculated according to monotonic transformation law and non-monotonic transformation law, respectively: For monotonic transformation law, we have:

Fig. 2 .

 2 Fig. 2. Phases of a porous material.

Fig. 3 .Fig. 4 .

 34 Fig. 3. Modification of pore radius due to salt precipitation and dissolution. = initial pore radius, = modified pore radius after salt precipitation and dissolution, = capillary pore radius at which waterfront is located, ( )= eroded size of pore.

  Fig.5billustrates non-monotonic transformation model whereby " ( ) is symmetrical S-shape function. Non-monotonic transformation restricts occurrence of salt precipitation and dissolution near waterfront in pores with radius betweenand + . Within the range of [ -, + ], modification on the pore radius increases from zero to maximum value, and reverts to zero. Pores located far from waterfront are barely affected by salt precipitation and dissolution such that modifications on these pores are disregarded. Therefore, non-monotonic transformation model is expressed as:

Fig. 5 .

 5 Fig. 5. Non-linear transformation models. (a) Monotonic transformation model and (b) Non-monotonic transformation model.

&

  is derived by integrating volume of pores with radius smaller than [

Fig. 6 .

 6 Fig. 6. Relationship between water saturation degree and location of waterfront.

Fig. 7 .

 7 Fig. 7. Pore size distribution of masonry cement. (a) Experimental and simulated cumulative volume fraction and (b) Incremental pore volume distribution.

Fig. 8 .

 8 Fig. 8. Profile for probability density curve of initial pore size distribution fitted by lognormal distribution.

Fig. 9 .

 9 Fig. 9. Linear transformation model of pore size distribution. (a) Probability density curve and (b) Cumulative volume fraction curve.

Fig. 10 .

 10 Fig. 10. Pore size distribution after single occurrence of salt precipitation. The initial porosity of 0.20 decreases to five degrees of porosity (0.19, 0.18, 0.17, 0.16, and 0.15) which corresponds to reductions in water saturation degree from 0.50 to five degrees (0.48, 0.46, 0.44, 0.42, 0.40).

Fig. 11 .

 11 Fig. 11. Pore size distribution after single occurrence of salt dissolution. The initial porosity of 0.20 increases to five degrees of porosity (0.21, 0.22, 0.23, 0.24, and 0.25) which corresponds to increases in water saturation degree from 0.40 to five degrees (0.42, 0.44, 0.46, 0.48, 0.50).

Fig. 12 .

 12 Fig. 12. Evolution of pore volume fraction due to salt precipitation and dissolution.

Fig. 13 .

 13 Fig. 13. Filling and emptying ratio after single salt precipitation and dissolution.

Fig. 14 .

 14 Fig. 14. Parameter ε against porosity for successive transformations.

Fig. 15 .

 15 Fig. 15. Effect of single and successive transformations on pore size distribution and filling/emptying ratio. Gradient color represents change in the pore size distribution and filling/emptying ratio.

Fig. 16 .

 16 Fig. 16. Evolution of water saturation degree due to successive transformations.

7 )

 7 For non-monotonic transformation law, 7 , 7 8 are defined as:

Table 1

 1 Analyzed properties of masonry cement and fitting values for parameters of lognormal distribution model.

	Matrix properties	Masonry cement
	Porosity	0.2029
	Cumulative pore volume (mL/g)	0.0971
	Specimen's diameter (mm)	35
	Specimen's thickness (mm)	11.10
	Mimimum radius (μm)	0.0015
	Maximum radius (μm)	160.2297
	Simulated	1.2
	Simulated	-1.8788
	Initial pore size distribution	See Fig.8

Table 2

 2 Values of parameters for successive transformations of pore size distribution.

	Chemical Reactions	Water Saturation Degree	Opening Zone	(μm)	Porosity
	Salt precipitation	0.4	0.1		0.20→0.15
		0.5
	Salt dissolution	0.4	0.1		0.20→0.25
		0.5

Acknowledgements

The author acknowledges Professor Ouali. Amiri and his research team, who provided experimental tested results for the initial pore size distribution of masonry cement. The author also thanks to all the reviewers for their insightful comments.

that the cumulative effects of successive transformations on pore size distribution should not be overlooked.

The proposed probabilistic model provides a reasonable interpretation on physical parameters and water-dependent evolution of pore size distribution. However, it is still not capable of accurately quantifying pore structure development in the course of salt precipitation and dissolution. The authors intend to study the proposed model by considering unstable waterfront and the environmental conditions that influence the waterfront. Most importantly, the model presented in this paper is purely theoretical. Although no appropriate experiments are available for the validation of the proposed model at present, it is necessary to make efforts to find out feasible experimental methods to validate the modeling results and further understand the mechanisms suggested by the proposed model.

Appendix A. Non-linear transformation models

The S-shape function for monotonic transformation law is expressed as:

For non-monotonic transformation law, we have:

G r J P r dr P r dr r r K r P r dr P r dr r


M.-P Noutary

N Biboulet

A A Lubrecht
email: ton.lubrecht@insa-lyon.fr

Mean pressure of a finite width Oil Control Ring with randomly spaced grooves

Keywords: OCR, surface texture variation, mean pressure

The current paper studies the pressure generated between two parallel grooved surfaces, as occurs in the oil control ring in internal combustion engines. It is shown that the pressure generated depends significantly on the position of the groove intersections, and thus that a single calculation cannot describe the pressure generated during a full passage of the ring over the pattern. Furthermore, it is shown that the introduction of a groove distance variation reduces the pressure variation, but does not affect the mean pressure generated.

INTRODUCTION

Internal combustion engine efficiency depends to a large extent on the friction in the Piston Ring Cylinder Liner contact. This contact has been studied numerically for more than four decades, see Dowson et al. [START_REF] Dowson | Piston ring lubrication part II: theoretical analysis of a single ring and a complete ring pack[END_REF] and Jeng [START_REF] Jeng | Theoretical Analysis of Piston-Ring Lubrication Part I -Fully Flooded Lubrication[END_REF][START_REF] Jeng | Theoretical Analysis of Piston-Ring Lubrication Part II -Starved Lubrication and the Application to the Complete Ring Pack[END_REF]. More general studies of the piston ring contact can be found in Tian et al. [START_REF] Tian | A Piston Ring-Pack Film Thickness and Friction Model for Multigrade Oils and Rough Surfaces[END_REF], Richardson [5], Gamble et al. [START_REF] Gamble | Detailed analysis of oil transport in the piston assembly of a gasoline engine[END_REF] and Tomanik [START_REF] Tomanik | Modelling the hydrodynamic support of cylinder bore and piston rings with laser textured surfaces[END_REF].

The third ring or the Oil Control Ring (OCR) has the particularity of pressure build-up being mostly dependent on the surface micro-geometry, as its surface is nominally flat and parallel to the liner. Pressure build-up in parallel textured surfaces has been studied by Etsion et al. [START_REF] Etsion | Analytical and experimental investigation of laser-textured mechanical seal faces[END_REF][START_REF] Kligerman | Improving tribological performance of piston rings by partial surface texturing[END_REF][START_REF] Etsion | Improving fuel efficiency with laser surface textured piston rings[END_REF], Tönder [START_REF] Tonder | Hydrodynamics effects of taylored inlet roughness: extended theory[END_REF] and Fowel et al. [START_REF] Fowell | Entrainment and inlet suction: two mechanisms of hydrodynamic lubrication in textured bearings[END_REF]. The importance of the cavitation pressure was demonstrated by Shen and Khonsari [START_REF] Shen | Effect of dimple's internal structure on hydrodynamic lubrication[END_REF][START_REF] Shen | Numerical optimization of texture shape for parallel surfaces under unidirectional and bidirectional sliding[END_REF][START_REF] Shen | The effect of laser machined pockets on the lubrication of piston ring prototypes[END_REF] and Ausas et al. [START_REF] Ausas | The impact of the cavitation model in the analysis of microtextured lubricated journal bearings[END_REF][START_REF] Ausas | Conservative one-dimensional finite volume discretization of a new cavitation model for piston-ring lubrication[END_REF]. An analysis on textured surfaces in parallel sliders was performed by [START_REF] Dobrica | Optimizing surface texture for hydrodynamic lubricated contacts using a mass-conserving numerical approach[END_REF][START_REF] Pascovici | Analytical investigation of a partially textured parallel slider[END_REF]. Algorithms stressing the importance of mass conservation were developped by Bertocchi et al. [START_REF] Bertocchi | Fluid film lubrication in the presence of cavitation: a mass-conserving two-dimensional formulation for compressible, piezoviscous and non-Newtonian fluids[END_REF], Profito et al. [START_REF] Profito | A General Finite Volume Method for the Solution of the Reynolds Lubrication Equation with a Mass-Conserving Cavitation Model[END_REF] and Woloszynski et al. [START_REF] Woloszynski | Efficient solution to the cavitation problem in hydrodynamic lubrication[END_REF]. Optimisation of the textured surface was performed by Codrignani et al. [START_REF] Codrignani | Optimization of surface textures in hydrodynamic lubrication through the adjoint method[END_REF]. Chunxing et al. [START_REF] Chunxing | Study on the mutual influence of surface roughness and texture features of rough-textured surfaces on the tribological properties[END_REF] and Wan et al. [START_REF] Ma | Cross-hatched groove influence on the load carrying capacity of parallel surfaces with random roughness[END_REF] analysed the interaction of pressure generated by grooves and surface roughness.

To improve the convergence of the mass-conserving algorithm by Woloszynski et al. [START_REF] Woloszynski | Efficient solution to the cavitation problem in hydrodynamic lubrication[END_REF] for very fragmented cavitated domains, a MultiGrid-like extension was implemented by Biboulet and Lubrecht [START_REF] Biboulet | Efficient solver implementation for Reynolds equation with massconserving cavitation[END_REF]. Noutary [START_REF] Noutary | A robust Reynolds solver for textured surfaces in the piston-ring cylinder liner contact[END_REF] studied different aspects of dimples and grooves in parallel surface lubrication, using a classical MultiGrid code.

Many parameters determine the pressure build-up in the OCR-liner contact, including micro texture parameters, such as groove depth, groove width, groove density, groove angle etc. Earlier work [START_REF] Noutary | Dimple influence on load carrying capacity of parallel surfaces[END_REF] has studied the influence of these parameters on friction and film generation. Perfectly periodic patterns were used and each calculation used a single set of groove depth, angle, width and distance. However, in reality these parameters are not deterministically known, but are part of a random distribution.

The current paper studies the influence of the groove pattern regularity in terms of mean pressure and mean pressure variation during crossing using a very robust and efficient Reynolds solver [START_REF] Biboulet | Efficient solver implementation for Reynolds equation with massconserving cavitation[END_REF]. Therefore, a groove pattern is generated with constant groove depth, width, angle and average groove distance. However, the real groove distance contains a random variation which makes the pattern no longer perfectly periodic. The random variation is a uniform distribution within a range maximum of ±20% of the nominal distance. Thus the groove distance varies inside the pattern, but more importantly, the position where the grooves intersect varies. In the rest of this paper, a periodic pattern (hence with no variation in the groove distance) will be designated as regular, conversely a non-periodic pattern will be designated as irregular.

EQUATIONS

The hydrodynamic lubrication of the OCR-cylinder liner contact can be described by the Reynolds equation. In order to have a mass conserving algorithm, the following complementary problem is solved:

∂ ∂x ( ρh 3 12η ∂p ∂x ) + ∂ ∂y ( ρh 3 12η ∂p ∂y )-u m ∂(ρθh) ∂x - ∂(ρθh) ∂t = 0 (1) with 1 (p > p cav if θ = 1) (p = p cav if 0 < θ < 1) (2)
where p is the hydrodynamic pressure, h the geometry, u m the mean velocity. The lubricant density ρ and the viscosity η are considered constant. The ring circumference is roughly 20 cm, and its width is 1.2 mm. However, such a large difference in dimensions is not practical from a calculation time point of view because the domain would require a very large number of points in the circumferential direction. Therefore, the calculation domain was chosen as a 24x1.2 mm rectangle, using Dirichlet boundary conditions. For a regular groove pattern, with a zero random distance, the problem can be modeled periodically in the circumferential direction and periodic boundary conditions could have been used there. However, with the random component added to the groove distance, this is no longer possible and Dirichlet conditions are required. These Dirichlet boundary conditions have a significant influence on the pressure distribution close to domain edges at y = ±12mm. Hence, only the central part of the domain y = ±6mm is used to compute the mean pressure in order to limit the influence of the Dirichlet boundary conditions. The pressure distribution depends on the exact position of the ring with respect to the groove pattern. Figure 1 shows that the maximum pressure reached in each pattern varies with its location.

Depending on which surface is textured, ring or liner, the problem is respectively steady-state or time dependent. However, considering an irregular pattern on the ring, its precise location with respect to the ring edges changes along the circumferential direction. To calculate an average value, one can either use a very large domain along the circumferential direction, or can perform several calculations shifting the pattern along the sliding direction. The latter option is chosen. If the ring is smooth and the pattern is on the liner, not studied in this paper, the calculation is transient and a time average can be calculated. In both cases, the mean pressure varies with the texture location. However, this paper is focused on the averaged value. The minimum mean pressure occurs when the groove intersections occur at the inlet edge. In this case the mean inlet height is the smallest, generating the smallest mean pressure, see the analogy with the Rayleigh step bearing in [START_REF] Noutary | Dimple influence on load carrying capacity of parallel surfaces[END_REF]. The steady-state assumption which is used means that one focuses on the central section of the piston stroke. The squeeze (positive or negative) which becomes significant close to the top and bottom dead center is thus neglected here.

For irregular patterns (see figure 1, operating conditions defined in table 1), one of the main conclusions of this work is that the variation amplitude diminishes in comparison with the regular pattern. In order to obtain an accurate estimation of the position-averaged mean pressure for all possible relative pattern positions during crossing, 16 calculations were performed with the flat ring "sliding" (shifting) over the groove pattern. The mean pressure and its standard deviation (among these 16 positions) are given in the following section. The operating conditions used throughout the paper are given in table 1, unless otherwise specified, the domain size is x = 1.2 × y = 24 mm, x is in the direction of movement. The groove angle is defined in the appendix fig [START_REF] Tonder | Hydrodynamics effects of taylored inlet roughness: extended theory[END_REF] The calculations were performed for a fixed minimum film thickness of 1µm. This choice means that the ring tension is not imposed through a force balance equation; the calculation results in a load carrying capacity expressed as the mean pressure.

RESULTS

Figure 2 shows the grooves in the central zone (left) and the corresponding generated pressure distribution (right). Please note that scale of the horizontal and vertical axis are different. For a regular groove pattern, all the grooves cross at the same position, hence the mean pressure strongly depends on the exact position of the groove crossings with respect to the ring position. Figure 3 shows the mean pressure for a regular pattern as a function of the relative position between pattern and ring (16 markers). It can be observed that the mean pressure varies roughly by a factor of 4 between the lowest and the highest value. Furthermore, as the pattern is regular, it defines the 16 positions x 0 of two full periods. These 16 positions will also be used for irregular patterns, where full periodicity ceases to exist. It will be shown that these 16 positions are sufficient to accurately describe the position averaged mean pressure. In order to make the groove pattern more realistic, a relative random component is added to the groove distance, all other groove parameters are fixed. The distance dist i between groove i and groove i + 1 is given by the following equation:

dist i = dist • (1.0 + var • rand i ) ( 3
)
where one draws from a uniform random distribution rand i ∈ [-1, +1], dist is the nominal groove distance (measured perpendicularly to the grooves) and var is the relative distance variation. In this paper, the variations used are: var = 0.0, 0.05, 0.1, 0. Figure 4 shows the position-averaged mean pressure (markers) and its standard deviation among the 16 positions (bars) for a regular pattern as a function of the nominal distance. For small nominal distance values, the positionaveraged mean pressure increases with distance, as the small groove distance limits pressure build-up. For a nominal distance of roughly 1 mm, the position-averaged mean pressure reaches a maximum. For even larger nominal distances, the position-averaged mean pressure diminishes. This can seem contradictory to the results from [START_REF] Biboulet | Cross hatched texture influence on the load carrying capacity of oil control rings[END_REF] where the mean pressure always increases with the groove distance. In this work the domain covered a complete pattern, therefore as the groove distance increased, the computational domain increased as well. However, in the current work, the physical domain width is limited to 1.2 mm, meaning that when the nominal groove distance increases, fewer grooves are included in the domain.

Thus when the nominal distance increases, parts of the domain do not contain a single groove and therefore the pressure over these parts is close to ambient, see figure 5. For the largest distance used in this work (41), the mean pressure is zero for 6 out of the 16 positions, as there is not a single groove inside the domain. Therefore it is expected that the position-averaged mean pressure is inversely proportional to the groove distance. In order for the reader to approximate the slopes in figure 4 Figure 6 shows the mean pressure as a function of the relative position x 0 between pattern and ring for var = 0.1 (16 positions). Compared with figure 3 where var = 0.0, the pressure variation is significantly reduced to ±10%. This figure also shows that the mean pressure is no longer perfectly periodic, the second maximum being significantly lower then the first one. Figure 7 shows the position-averaged mean pressure as a function of the groove distance for a relative distance variation of 10% (var = 0.1). The position-averaged mean pressure values are very similar to the ones in figure 4. However, for small nominal distances, the standard deviation (bars) is much smaller, as now the groove distance variations cause the groove intersections to be evenly distributed over the entire domain, including the inlet and outlet. For large nominal distances, the standard deviation increases again, as part of the domain lacks grooves, and its pressure is thus equal to the ambient pressure. For these large groove distances, the standard deviation in the mean pressure is similar to the one in figure 4.

For all calculations, the variations in mean pressure would have been smaller if the domain in the y dimension would have been larger, as one averages over a larger sample. That is why this paper primarily focuses on the position-averaged mean pressure. Figure 8 shows the results with a 20% variation in the nominal groove distance (var = 0.2). The mean pressure and its standard deviation behave in a very similar manner to figure 7 where var = 0.1. In order to validate the accuracy of the trends of the position-averaged mean pressure as a function of the groove mean distance, three different random series were used, and the results are plotted in Figure 9. The differences are minor over the entire distance range, but smallest around dist = 1. This means that our samples are sufficiently large to limit the mean pressure variation from one ensemble to another, i.e. the domain size and the value of 16 x 0 positions is sufficient.

Figure 9: position-averaged mean pressure as a function of average groove distance, for three different series of random numbers, var = 0.2.

To complete our understanding of the origins of the position-averaged mean pressure variations as a function of the random distance component, the mean pressures for var ∈ [0.0, 0.05, 0.1, 0.2] are plotted in Figure 10. For small distances, the variations are minor. For distances between 1 and 10, the variation are a little larger. For dist ≥ 10 the absolute variations are very small again. A careful observer will only spot 3 dots instead of four. The results for var = 0 and var = 0.05 differed by less then 2 percent, and the two symbols overlap. The fact the position averaged mean pressure depends very little on the groove distance variation var, indicates that the pressure build-up is determined by the local geometry and that the neighbouring geometry plays only a small role. As such averaging the mean pressure will always give a very similar value. Hence the random distance component tends to decrease the mean pressure variation as a function of position, but hardly affects the average mean pressure. Another way to explain this result is that, since the distance variation around the nominal value is limited, one can perform a local linearisation around the nominal value, and patterns with a contribution below average will be compensated by patterns above the average.

As such it is sufficient to study the mean pressure generated by a groove pattern over 16 positions, and knowing that this mean pressure will be very close to that generated by any random distance distribution with the same mean distance.

Typical oil control rings have a smaller land width (0.15 to 0.4 mm) than the 1.2 mm chosen in this study. However, the presented results are useful for an entire range of widths. The implication of a smaller width is that for a given groove distance, the transition from the left to the right part of fig 10 will appear sooner. A smaller size means that a domain without a single groove occurs more rapidly. This transition would occur around the dimensionless groove distance (groove distance over ring width) of 0.8. An other way to see this limit is that the pattern size along the sliding direction must not be much larger than the ring width. The optimum value of the groove distance should be considered relatively to the ring width. If the pattern size is sufficiently small in order to remain on the left part of fig 10, the pressure distribution is well spread on every plateau such as the case illustrated in fig 2 . Obviously, this transition is most likely dependent on the groove angle too. Finally, the main conclusions of this study are unchanged: first, the calculation of a single pattern position is not sufficient to judge the performance of a texture, second, the position averaged mean pressure is not modified by a random variation of the groove distance.

CONCLUSION

The current paper shows the influence of groove distance on the load carrying capacity (mean pressure) for a flat ring (Oil Control Ring). The texture performance can not be predicted based on a single texture location. It was also shown that for small inter groove distances, the mean pressure increases with the inter groove distance. For a distance slightly lower than the ring width, a maximum is reached. For even larger nominal distances the mean pressure diminishes. As the groove pattern has a similar dimension as the ring, the precise position of the ring with respect to the pattern greatly influences the mean pressure built-up. A significant standard deviation for different pattern locations is observed for small distances. When a random variation is added to the groove distance, this large standard deviation decreases for small inter-groove distances. For large inter-groove distances the standard deviation remains large for all cases but this last point is most likely domain size dependent. For these large inter-groove distances, the mean pressure fluctuation is caused by the grooves appearing in the domain or not at all and not by the position of the groove intersections.

Finally, the position-averaged mean pressure depends very little on the variation of the groove distance, hence only the mean groove distance is a significant parameter for pressure generation.

Figure 1 :

 1 Figure 1: Full calculational domain x ∈ [0, 1.2] mm, y ∈ [-12, 12] mm with an irregular groove pattern, geometry (left) geometry colour bar in µm and pressure (right) pressure colour bar in Pa

Figure 2 :

 2 Figure 2: Central part of domain for an irregular pattern, groove geometry (left), pressure (right) (dist = 0.42 mm, var = 20%, geometry colour bar µm, pressure colour bar in Pa

Figure 3 :

 3 Figure 3: Mean pressure for a regular pattern as a function of the ring position x 0 (shift), dist = 0.14mm

  2. A value of var = 0.0 corresponds to a regular pattern. Different seeds are used to obtain different random series.

Figure 4 :

 4 Figure 4: Position-averaged mean pressure (markers) as a function of average groove distance for a regular pattern var = 0.0.

  Figure4shows the position-averaged mean pressure (markers) and its standard deviation among the 16 positions (bars) for a regular pattern as a function of the nominal distance. For small nominal distance values, the positionaveraged mean pressure increases with distance, as the small groove distance limits pressure build-up. For a nominal distance of roughly 1 mm, the position-averaged mean pressure reaches a maximum. For even larger nominal distances, the position-averaged mean pressure diminishes. This can seem contradictory to the results from[START_REF] Biboulet | Cross hatched texture influence on the load carrying capacity of oil control rings[END_REF] where the mean pressure always increases with the groove distance. In this work the domain covered a complete pattern, therefore as the groove distance increased, the computational domain increased as well. However, in the current work, the physical domain width is limited to 1.2 mm, meaning that when the nominal groove distance increases, fewer grooves are included in the domain.Thus when the nominal distance increases, parts of the domain do not contain a single groove and therefore the pressure over these parts is close to ambient, see figure5. For the largest distance used in this work (41), the mean pressure is zero for 6 out of the 16 positions, as there is not a single groove inside the domain. Therefore it is expected that the position-averaged mean pressure is inversely proportional to the groove distance. In order for the reader to approximate the slopes in figure4, two solid curves are drawn on fig 4: p m ∝ x 2 (left) and p m ∝ 1/x (right).

Figure 5 :

 5 Figure 5: Groove geometry and pressure distribution for a large nominal groove distance of dist = 10 and var = 0.2, central part of the domain.

Figure 6 :

 6 Figure 6: mean pressure as a function of the ring position x 0 with respect to the groove pattern var = 0.1, dist = 0.14mm.

Figure 7 :

 7 Figure 7: position-averaged mean pressure as a function of average groove distance var = 0.1.

Figure 8 :

 8 Figure 8: position-averaged mean pressure as a function of average groove distance var = 0.2.

Figure 10 :

 10 Figure 10: position-averaged mean pressure as a function of average groove distance, var ∈ [0.0, 0.05, 0.1, 0.2].

Table 1 :

 1 . Operating conditions and geometry

	η 0	0.01 Pa s
	u m	0.5 m/s
	p amb	30 kPa
	angle	60 •
	groove depth 1.0 µm
	groove width 0.1 mm

A Calculation window

In this section, the choice of the calculation window (orange zone in Figure 11) is explained. One regular pattern is considered. The calculation zone is a 24 × 1.2 mm band starting in the sliding direction at x 0 and ending at x 1 . The segment AB is decomposed into sixteen equally spaced segments and the point x 0 is used to define the 16 positions of the calculation window. The total useful domain in the sliding direction is then the segment AC of total length d/ sin α + 1.2 mm where d denotes the nominal distance between grooves and α, the groove angle. In Figure 11, only the center line of the grooves is plotted. Of course, when d is much larger than 1.2 mm, as it is the case in the figure, there are only a few grooves inside the calculation zone.