Sampling

A randomly selected batch of 500 one-piece natural cork stoppers of the first commercial quality and nominal dimensions of 24x44 mm were sent to the INIA-CIFOR cork laboratory from a Spanish cork stopper manufacturer. A sample of 35 cork stoppers selected with the criteria of having similar values of density were used. This criteria was applied to try to play down the influence of stopper density in its mechanical properties (Anjos et al., 2008; Anjos et al., 2014; Rosa & Fortes, 1988). The sample of 35 natural cork stoppers were randomly subsampled in 7 groups with 5 stoppers per group, corking each group using different diameters of the tube that simulate bottleneck. Those diameters were 17, 18, 18.4, 19, 20, 21 and 22 mm.

Cork stopper were acclimatized at 20 °C and 65 % of relative humidity. Stabilization was considered to have been achieved when the weight variation in two consecutive weighings was less than 0.024 g (which is equivalent to a humidity difference of 0.1 %). Once acclimatized, stoppers were weighed and measured using Mitutoyo ID-F150 digital vernier calipers. Room conditions during the measures were an ambient temperature of 20 ºC ± 4 ºC and relative humidity of 50% ± 10%. Stoppers density was calculated as already reported in González-Hernández et al. (2014).

Mechanical tests

The biaxial compression force (Fc) in the modelling sample set was measured using a semiautomatic corking machine equipped with a load cell (UTILCELL, Mdo: 650 SNo 460775(02) Emax: 2Tn), as already reported in González-Hernández et al. (2014). Cork stoppers were inserted in each bottleneck tube by applying the same compression rate of 33 %.

The diameter recovery evolution of each cork stopper was assessed by measuring the stopper diameter 24 hours after the test with Mitutoyo ID-F150 digital vernier callipers.

The relaxation force (Fr) (González-Hernández et al, 2014) in the both sample set was measured by inserting each bottleneck tube with the stopper inserted within it into the device developed in the INIA-CIFOR Cork Laboratory (González-Hernández et al, 2012) as already reported in González-Hernández et al. (2014).

Statistical analysis

For each measured variable, the mean, standard deviations, minimum and maximum were calculated. In a preliminary analysis, the assumptions of normality, independence and homogeneity of the variance were verified. The hypothesis of no differences among stopper included in each subsampled group was tested using ANOVA. The differences were examined using pairwise comparisons according to the Tuckey test. All tests were conducted at the α=0.05 level and all analyses were carried out using the SAS software version 9.4 (SAS Institute Inc., 2016).

Theoretical model

The strain-stress curves described for cork by Fortes et al. (2004) and Mano (2002) are very similar to those that take place during corking. In the formers, the compression load is done in a unique axial direction, while during corking the cork stopper is subjected to a concentric compression. Fig. 1 shows a schematic depiction of the evolution of the stressstrain hysteresis loop during corking, sealing period and uncorking.

Figure 1. Schematic depiction of the evolution of the stress–strain hysteresis loop during corking, sealing period and uncorking. Ds: Stopper diameter. Dr: Recovered diameter. Db: Bottleneck diameter. Dg: Caliper diameter of the corking machine. Fc: Maximum compression force. F1: Compression force when the cork stopper is ftted in the bottleneck.

During corking, the cork stopper diameter is compressed from its initial value (Ds) to the caliper diameter of the corking machine (Dg). The stress applied increases to a maximum value of compression (Fc, point C in Fig. 1) describing the loading curve 0C. At point C, when the cork stopper is fitted in the bottleneck of diameter (Db), the stress is reduce from the compression force (Fc, point C in Fig. 1) to the relaxation force (Fr, point H in Fig. 1) and simultaneously the diameter of the cork stopper is partly recovered from Dg to Db, describing the unloading curve CH. When the cork stopper is fitted in the bottleneck the diametric recovery goes on describing the whole unloading curve until reaching the maximum value of the recovered diameter (Dr) that the stopper will reach as long as the wine is bottled.

The similarity of triangles ACF and ABE allows getting to:

Therefore,

For a given value of bottleneck diameter (Db), the values of Fc and F1 can be related through a coefficient

Replacing k in (2) we get to:

If we take into account that the recovered diameter (Dr) can also be expressed as a function of the diameter recovery (RD) and the caliper closure diameter can also be expressed as a function of the compression rate (CR). Then equation (4) can be expressed as a function of those variables:

Where Fr is the relaxation force; Fc is the compression force; k is the relaxation coefficient, Ds is the stopper diameter, Db is the bottleneck diameter, Dr is the recovered diameter, Dg is the caliper closure diameter, RD is the diameter recovery and CR is the compression rate. The relationship between Fr and Fc can be expressed by means of the relaxation ratio (RR) (González-Hernández et al. 2014):

Then,

Where k values depends on the quality of cork stoppers, the compression rate applied when corking, the bottleneck diameter and the stopper diameter. If we were able to determine which values should have the coefficient k for a proper sealing when using cork stoppers of a given quality, applying the same compression rate and using different bottlenecks diameter, we could establish the proper stopper diameter for each bottleneck diameter by applying the following equation obtained from equations (5) and (6):

The stopper diameter depends mainly on the bottleneck diameter (Db), the compression rate (CR) and the stopper quality, because the relaxation ratio (RR) and the diametric recovery (RD) depends on stopper quality and the compression rate applied in the bottling operation (Anjos et al., 2008; Fortes et al., 2004; González-Hernández et al., 2014).

Optimum stopper diameter when reducing bottleneck diameter

The statistics values for each measured variable are shown in Table 1. ANOVA indicated no significant differences in biaxial compression force (Fc) values among subsample groups (F value = 1.92 p value = 0.1772). This result was expected because the same compression rate was used. The compression force takes mean values of 228.91 daN, between 193.80 and 261.00 daN (Table 1). These values are very similar to those obtained by previous studies (Beorlegui, 2014; González-Hernández et al., 2014; Prades et al., 2014) for natural cork stopper of the same quality which is expected since measurement was taken in the same way using a clamp or caliper which means that are values of biaxial compression.

The maximum relaxation force (Fr) applied by each stopper against the bottleneck tube, and measured using the device developed in the INIA-CIFOR Cork Laboratory (González-Hernández et al., 2012), shows significant differences among subsampled groups of stoppers (Table 1 and Fig. 2) except for 18.4 mm which mean values is not significantly different from the man value for 18 mm and 19 mm. As expected, the relaxation force decrease as the bottleneck diameter increase when using natural cork stoppers of 22 mm of diameter.

Figure 2. Boxplots of relaxation force distributions per subsampled group by bottleneck diameter. Boxplot notches indicate a 95 % confdence interval on the median. Different letters show signifcant differences (p<0.05).

The diametric recovery (RD) after 24h from compression do not show significant differences among subsampled groups of stoppers. Since the same compression rate was applied in all of them, this result was also expected. The mean value for the whole modelling sampled set was 95.929 % ranging from 94.539 % to 97.433 %. Then, it can be considered that after 24h after compression to 33 % the tested natural cork stoppers recovered 96 % of its initial diameter. In Fig. 1, Dr refers to the value of the recovered diameter that cork stopper would reach 24 h after being fitted in the bottleneck, we are going to consider that this diameter is the 96 % of the stopper diameter. Following this assumption, we have calculated k values for the modelling sampled set (Table 1).

The relationship between bottleneck diameter and k follows a parabolic pattern (Fig. 3). The minimum point of the second degree parabola is reached at a bottleneck value of 19.2 mm, all the values of k above the minimum value corresponding to bottleneck dimeters smaller than 19.2 mm, can be considered as safe from a sealing performance perspective when the compression rate applied is 33 %. The equation shows in Fig. 3 jointly to equation (8) can be applied to estimate stopper diameter for a given bottleneck diameter smaller than 19 mm considering a compression rate of 33 % and, as relaxation ratio value, the mean value obtained for the sampled group of cork stoppers fitted in bottleneck tubes of 19 mm. Therefore, for a bottleneck diameter of 18 mm the value of the stopper diameter should be at least of 22.3 mm, and for a bottleneck diameter of 17 mm the value of the stopper diameter should be at least of 20.3 mm.

Figure 3. Variation of the k values calculated in the modelling sampled set with bottleneck diameter. Solid line represent the regression line. Dark-shaded region shows the 95 % confdence interval and dashed lines are upper and lower 95 % prediction intervals of the regression.