Biogenic oligomers as reactive additives for the curing of reactive resins

Abstract

A reactive resin includes a vinyl ester resin as a base resin and an oligomeric itaconic acid ester as a reactive diluent.

Claims

1: A reactive resin, comprising: i) a base resin comprising at least one vinyl ester resin, and ii) at least one itaconic acid ester of formula (I), ##STR00006## in which R represents hydrogen or a C.sub.1-C.sub.6 alkyl group, X represents a C.sub.2-C.sub.10 alkylene group, and n is 2.

2: The reactive resin according to claim 1, wherein the at least one itaconic acid ester of the formula (I) has a weight-average molar mass M.sub.w of at least 500 g/mol.

3: The reactive resin according to claim 1, wherein the at least one itaconic acid ester of the formula (I) is completely obtainable from a renewable raw material.

4: The reactive resin according to claim 1, wherein the at least one vinyl ester resin is a vinyl urethane ester resin.

5: The reactive resin according to claim 1, further comprising at least one inhibitor.

6: The reactive resin according to claim 5, wherein the at least one inhibitor is selected from the group consisting of 2-methoxyphenol, 4-methoxyphenol, 2,6-di-tert-butyl-4-methylphenol, 2,4-di-tert-butylphenol, 2,6-di-tert-butylphenol, 2,4,6-trimethylphenol, 2,4,6-tris(dimethylaminomethyl)phenol, 4,4-thio-bis(3-methyl-6-tert-butylphenol), 4,4-isopropylidenediphenol, 6,6-di-tert-butyl-4,4-bis (2,6-di-tert-butylphenol), 1,3,5-trimethyl-2,4,6-tris(3,5-di-tert-butyl-4-hydroxybenzyl)benzene, 2,2-methylene-di-p-cresol, pyrocatechol, 4-tert-butyl pyrocatechol, 4,6-di-tert-butyl pyrocatechol, hydroquinone, 2-methylhydroquinone, 2-tert-butylhydroquinone, 2,5-di-tert-butylhydroquinone, 2,6-di-tert-butylhydroquinone, 2,6-dimethylhydroquinone, 2,3,5-trimethylhydroquinone, benzoquinone, 2,3,5,6-tetrachloro-1,4-benzoquinone, methylbenzoquinone, 2,6-dimethylbenzoquinone, naphthoquinone, and mixtures thereof.

7: The reactive resin according to claim 5, comprising: i) 50.0 to 95.0 wt. % of the base resin, comprising the at least one vinyl ester resin, ii) 5.0 to 40.0 wt. % of the at least one itaconic acid ester of the formula (I), and iii) up to 1.0 wt. % of the at least one inhibitor, based on a total weight of the reactive resin.

8: A reactive resin component comprising: the reactive resin according to claim 1, and at least one inorganic or organic aggregate.

9: The reactive resin component according to claim 8, wherein the at least one inorganic or organic aggregate is selected from the group consisting of fillers, thickeners, thixotropic agents, non-reactive solvents, agents for improving flowability, wetting agents, and mixtures thereof.

10: The reactive resin component according to claim 8, comprising: 30 to 80 wt. % of the at least one inorganic or organic aggregate, based on a total weight of the reactive resin component.

11: A multi-component system, comprising: A) the reactive resin component according to claim 8, and B) a hardener component.

12: The multi-component system according to claim 11, wherein the hardener component comprises: a radical initiator as a curing agent, and optionally inorganic and/or organic aggregates.

13: The multi-component system according to claim 11, wherein a weight ratio of the reactive resin component to the hardener component is in a range of approximately 3:1 to approximately 7:1.

14: A method for chemical fastening, comprising: fastening with the reactive resin of claim 1.

15: The method according to claim 14, wherein the chemical fastening is fastening of an anchor in a borehole.

16: The method according to claim 15, wherein the anchor comprises steel or iron.

17: The method according to claim 14, wherein the reactive resin further comprises: a base resin comprising at least one vinyl ester resin, and at least one inhibitor.

18: The method according to claim 17, wherein the at least one vinyl ester resin is a vinyl urethane ester resin.

19: The method according to claim 17, wherein the at least one inhibitor is selected from the group consisting of 2-methoxyphenol, 4-methoxyphenol, 2,6-di-tert-butyl-4-methylphenol, 2,4-di-tert-butylphenol, 2,6-di-tert-butylphenol, 2,4,6-trimethylphenol, 2,4,6-tris(dimethylaminomethyl)phenol, 4,4-thio-bis(3-methyl-6-tert-butylphenol), 4,4-isopropylidenediphenol, 6,6-di-tert-butyl-4,4-bis (2,6-di-tert-butylphenol), 1,3,5-trimethyl-2,4,6-tris(3,5-di-tert-butyl-4-hydroxybenzyl)benzene, 2,2-methylene-di-p-cresol, pyrocatechol, 4-tert-butyl pyrocatechol, 4,6-di-tert-butyl pyrocatechol, hydroquinone, 2-methylhydroquinone, 2-tert-butylhydroquinone, 2,5-di-tert-butylhydroquinone, 2,6-di-tert-butylhydroquinone, 2,6-dimethylhydroquinone, 2,3,5-trimethylhydroquinone, benzoquinone, 2,3,5,6-tetrachloro-1,4-benzoquinone, methylbenzoquinone, 2,6-dimethylbenzoquinone, naphthoquinone, and mixtures thereof.

20: The method according to claim 17, wherein the reactive resin comprises: i) 50.0 to 95.0 wt. % of the base resin, comprising the at least one vinyl ester resin, ii) 5.0 to 40.0 wt. % of the at least one itaconic acid ester of formula (I), and iii) up to 1.0 wt. % of the at least one inhibitor, based on the total weight of the reactive resin.

Description

PRACTICAL EXAMPLES

[0101] 1. Preparation of the Oligomeric Itaconic Acid Esters

[0102] The reaction vessel (RC-1, Mettler Toledo), preheated to approximately 50 C., was stabilized with 1012.16 g (6.4 mol) dimethyl itaconate (DMI; TCI>98%, stabilized with hydroquinone monomethyl ether (HOME)), 504.56 g (5.6 mol) 1,4-butanediol (Aldrich, 99%) and 0.72 g HQME (SIGMA Aldrich, ReagentPlus 99%, 0.103 mol. % based on DMI) and the mixture is homogenized with stirring. Subsequently, 10.116 g (1 wt. % based on DMI) Ti(OBu).sub.4 (Aldrich, 97%) was added. After the apparatus was closed, gradual heating to 150 C. was carried out. The methanol released from the apparatus began to distill from a melt temperature of approximately 125 C. It was condensed and collected in the mounted Liebig cooler. After the start of the reaction, the temperature was permanently adjusted in order to ensure that the methanol released was distilled off uniformly. The progress of the reaction was monitored by .sup.1H-NMR. The transesterification was completed a maximum of 5 hours after the distillation of the MeOH had started and the composition was cooled to approximately 50 C. After reaching this temperature, a vacuum (max. 20 mbar) was applied to remove the remaining methanol released. After a further 4 h, the residual MeOH content was reduced to <0.3 wt. %.

[0103] The molar mass (M.sub.w) of the itaconic acid esters obtained was determined as follows: the MALDI tests were carried out using an Autoflex Speed TOF/TOF system (Bruker Daltonics GmbH) using a pulsed laser beam at an acceleration voltage of 20 kV in the reflector or linear mode. For the preparation, the oligomers, the matrix dithranol and the salt sodium trifluoroacetate were dissolved in chloroform, mixed and dropped onto a target. The measurement was carried out after the solvent had evaporated. FIG. 1 shows the results of the molar mass determination.

[0104] The product could be left out and used without further processing for use in reactive resins or reactive mortars. Table 1 lists the results of a total of 6 tests to illustrate the reproducibility of the transesterification. Comparable products were obtained.

TABLE-US-00001 TABLE 1 Results of the transesterification tests Content of Residual Conversion double-esterified content of OCH3 butanediol butanediol Sample [mol. %] [mol. %] [mol. %] A.sup.1) 77 75 2 B.sup.1) 74 69 3 C.sup.1) 75 73 2 D.sup.1) 75 71 2 E.sup.1) 77 77 2 F.sup.2) 76 75 2 .sup.1)Carried out in the 1.8 L reactor .sup.2)Carried out in the 500 mL reactor

[0105] 2. Investigation of the Curing Behavior

[0106] The oligomeric itaconic acid ester obtained in Example 1 was added to reactive resins and their curing behavior was then examined. A mixture of urethane methacrylate resin (master batch A1), hydroxypropyl methacrylate (HPMA), the commercial reactive diluent 1,4-butanediol dimethacrylate (1,4-BDDMA) an aromatic amine (as an accelerator for peroxide decomposition) and TEMPOL and tert-butyl pyrocatechol (tBBK) was used as the standard resin. In this reactive resin, different amounts of the reactive diluent 1,4-BDDMA were replaced by the itaconic acid esters (Sample F) prepared in Example 1 (see Table 2). For curing, the reactive resin was mixed with benzoyl peroxide (Perkadox 20S, Akzo Nobel) in a suitable ratio.

TABLE-US-00002 TABLE 2 Composition of the investigated reactive resins Proportion of itaconic acid Itaconic acid ester A1 1,4-BDDMA ester DiPT TEMPOL tBBK Sample [mol. %] [g] [g] [g] [g] [g] [g] Reference 0% 42.68 25.60 0 1.47 0.02 0.22 Sample 1 20% 42.68 20.48 7.76 1.47 0.02 0.22 Sample 2 40% 39.70 14.29 14.42 1.37 0.02 0.21 Sample 3 60% 38.37 9.20 20.9 1.32 0.02 0.20 Sample 4 80% 37.11 4.45 26.94 1.28 0.02 0.20

[0107] The temperature-time curve of the curing was then recorded as follows: Approximately 20 g of the reactive resin to be examined and the corresponding amount of hardener (Perkadox 20S, weight ratio 70:30) were weighed out in a plastic beaker. As the system is sensitive to the ambient temperature, the components must be kept at 25 C. The temperature was controlled in a thermostat (B12/C11 Prfgertewerk Medingen GmbH). The measurement was started immediately before the reaction components were mixed. The hardener was added to the resin component and stirred well with a wooden spatula for 40 s. The mixture was poured into two test tubes approximately 6 cm high, each of which was suspended separately in a measuring cylinder located in the thermostat. A temperature sensor (K-type, 150 mm long 1.5 mm) coated with silicone paste was then immersed in the middle of each mixture at a depth of 2 cm. Since the ambient temperature was registered until the sensors were immersed, the shape of the curve at the start of the measurement is not relevant, which is why the temperature-time curves were only used for the evaluation from 100 seconds. The temperature curve was registered by means of the sensors connected to a Voltkraft Datalogger K202 (connected to a PC). The maximum temperature of the curve (T.sub.max) and the time at 35 C. were read off as results in the shape of the curve (schematically shown in FIG. 2). Three duplicate determinations were made per system. The measured temperature-time curves are shown in FIG. 3.

[0108] The maximum temperature of the composition T and the time taken to reach this temperature T.sub.max were evaluated as the results of these measurements. A T.sub.max (a measure of the heat of polymerization released during curing) that is comparable to the reference indicates the desired incorporation of the added reaction products into the network being formed. The percentages given in the following for the addition of the oligomeric itaconic acid ester in mol. % are based on the proportion of 1,4-BDDMA in the mixture. The number of double bonds in the itaconic acid ester is taken into account in these calculations, so that there is always an approximately constant amount of reactive double bonds in the mixture. The results are summarized in Table 3.

[0109] As the proportion of itaconic acid esters in the reactive resin increases, the T.sub.max drops to approximately 130 C., while the times until the T.sub.max is adjusted decrease. The results show that the 1,4-BDDMA can be replaced by the itaconic acid ester without this having a negative effect on the curing reaction.

TABLE-US-00003 TABLE 3 Results of the curing tests mol. % 1,4-BDDMA replaced by T.sub.max Tmax Sample itaconic acid ester [ C.] [ C.] [min] [min] Reference 0 156 1 05:06 00:05 Sample 1 20 153 1 04:05 00:08 Sample 2 40 146 1 03:39 00:04 Sample 3 60 137 2 03:21 00:04 Sample 4 80 128 1 03:14 00:05

[0110] 3. Preparation of Reactive Resin Systems

[0111] A1: Reactive Resin Masterbatch A1 was Prepared in the Following Way:

[0112] The reactive resin master batch was synthesized with 65 wt. % of the comparative compound 1 as the base resin and 35 wt. % hydroxypropyl methacrylate (Visiomer HPMA; Evonik Degussa GmbH), in each case based on the total weight of the reactive resin master batch, according to the method in EP 0 713 015 A1, which is hereby introduced as a reference and reference is made to the entire disclosure thereof. The product has the following structure, there being an oligomer distribution where n=0 to 3:

##STR00004##

[0113] B1: Reactive Resin Masterbatch B1 was Prepared in the Following Way:

[0114] 80400 g of hydroxypropyl methacrylate (Visiomer HPMA; Evonik Degussa GmbH) were provided in a 300 liter steel reactor having an internal thermometer and stirrer shaft and were mixed with 36 g phenothiazine (D Prills; Allessa Chemie), 70 g 4-hydroxy-2,2,6,6-tetramethyl-piperdinyl-1-oxyl (TEMPOL; Evonik Degussa GmbH) and 56 g dioctyltin dilaurate (TIB KAT 216; TIB Chemicals). The batch was heated to 60 C. Subsequently, 69440 g of methylene di(phenyl isocyanate) (MDI; Lupranat MIS; BASF SE) were added dropwise with stirring for 1.5 h. The mixture was then stirred at 80 C. for a further 45 minutes. 50,000 g 1,4-butanediol dimethacrylate (Visiomer 1,4-BDDMA, Evonik Degussa GmbH) were then added. The reactive resin master batch B1 was obtained, which contains 75 wt. % of the compound shown below as a base resin and 25 wt. % 1,4-butanediol dimethacrylate (Visiomer 1,4-BDDMA, Evonik Degussa GmbH), based on the total weight of the reactive resin master batches. The compound has the following structure:

##STR00005##

[0115] A2: Reactive Resin Masterbatch A2 was Prepared from Reactive Resin Master Batch A1 in the Following Way:

[0116] 1147.8 g (57.34 wt. %) of master batch A1 was mixed with 400 g (20 wt. %) 1,4-butanediol dimethacrylate (Visiomer 1,4-BDDMA, Evonik Degussa GmbH), 46 g (2.3 wt. %) di-isopropanol-p-toluidine (BASF SE), 4.6 g (0.23 wt. %) catechol (Catechol Flakes, RHODIA) and 1 g (0.05 wt. %) tert-butyl pyrocatechol (tBBK, CFS EUROPE S.p.A. (Borregaard Italia S.p.A.)) and stirred until completely homogenized.

[0117] B2: Reactive Resin Masterbatch B2 was Prepared from Reactive Resin Master Batch B1 in the Following Way:

[0118] 1013.4 g (50.67 wt. %) of master batch B1 was mixed with 400 g (20 wt. %) 1,4-butanediol dimethacrylate (Visiomer 1,4-BDDMA, Evonik Degussa GmbH), 162.6 g (8.13 wt. %) hydroxypropyl methacrylate (Inchem), 22.4 g (1.12 wt. %) di-isopropanol-p-toluidine (BASF SE) and 0.3 g (0.015 wt. %) 4-hydroxy-2,2,6,6-tetramethyl-piperidinyl-1-oxyl (TEMPOL; Evonik Degussa GmbH) and stirred until completely homogenized.

[0119] The reactive resins A3.1, A3.2 and A3.3 and B3.1, B3.2 and B3.3 were then produced by the dissolving or mixing and subsequent homogenization of the reactive resin master batches A2 and B2, respectively. The compositions of the reactive resins are summarized in Tables 4a and 4b.

TABLE-US-00004 TABLE 4a Compositions of the reactive resins (A series) Reactive resin (total quantity) A3.1 A3.2 A3.3 (500 g) (400 g) (336 g) Master batch A2 [g] 399.6 319.7 268.2 [wt. %] 79.92 79.92 79.92 Catechol [g] 0.2 0.2 0.1 [wt. %] 0.04 0.05 0.03 tBBK [g] 0.2 0.2 0.1 [wt. %] 0.04 0.05 0.03 1, 4-BDDMA [g] 100 [wt. %] 20 Methyl butandiol itaconate [g] 80 [wt. %] 20 Diethyl malonate [g] 67.1 [wt. %] 20

TABLE-US-00005 TABLE 4a Compositions of the reactive resins (B series) Reactive resin (total quantity) B3.1 B3.2 B3.3 (336 g) (340 g) (340 g) Master batch B2 [g] 271.8 271.4 272.0 [wt. %] 79.94 79.94 79.94 tBBK [g] 0.34 0.41 0.22 [wt. %] 0.1 0.12 0.065 1,4-BDDMA [g] 68 [wt. %] 20 Methyl butandiol [g] 68 itaconate [wt. %] 20 Diethyl malonate [g] 68 [wt. %] 20

[0120] The reactive resin components A4.1, A4.2 and A4.3 and B4.1, B4.2 and B4.3 were prepared from the reactive resins A3.1, A3.2 and A3.3 and B3.1, B3.2 and B3.3, respectively, as follows:

[0121] The reactive resin was mixed with Secar80 (Kemeos Inc.), Cab-O-Sil TS-720 (Cabot Corporation), Aerosil R-812 (Evonik) and quartz sand F32 (Quarzwerke GmbH) in a dissolver under vacuum (the respective amounts can be found in Tables 3a and 3b below). Mixing took place with a PC laboratory system dissolver of the type LDV 0.3-1 for 8 minutes (2 min: 2500 rpm; then 6 min: 3500 rpm; each at a pressure <100 mbar) with a 55 mm dissolver disc and an edge scraper. The compositions of the reactive resin components are summarized in Tables 5a and 5b.

TABLE-US-00006 TABLE 5a Compositions of the reactive resin components (A series) Reactive resin component (total quantity) A4.1 A4.2 A4.3 (900 g) (900 g) (720 g) Reactive resin [g] 310.5 (A3.1) 310.5 (A3.2) 248.0 g (A3.3) [wt. %] 34.5 34.5 34.5 Secar 80 [g] 166.5 166.5 133.4 [wt. %] 18.5 18.5 18.5 Cab-O-Sil [g] 9.0 9.0 7.2 TS-720 [wt. %] 1.0 1.0 1.0 Aerosil [g] 16.2 16.2 13.0 R-812 [wt. %] 1.8 1.8 1.8 Quartz sand [g] 398 398 318.2 F32 [wt. %] 44.2 44.2 44.2

TABLE-US-00007 TABLE 5b Compositions of the reactive resin components (B series) Reactive resin component (total quantity) B4.1 B4.2 B4.3 (840 g) (840 g) (835 g) Reactive resin [g] 289.9 (B3.1) 289.9 (B3.2) 289 (B3.3) [wt. %] 34.5 34.5 34.5 Secar 80 [g] 155.3 155.3 154.6 [wt. %] 18.5 18.5 18.5 Cab-O-Sil [g] 8.9 8.4 8.3 TS-720 [wt. %] 1.0 1.0 1.0 Aerosil [g] 15.2 15.1 15.0 R-812 [wt. %] 1.8 1.8 1.8 Quartz sand [g] 371.2 371.0 370.0 F32 [wt. %] 44.2 44.2 44.2

[0122] The two-component reactive resin systems A5.1, A5.2 and A5.3 were then prepared from the reactive resin components A4.1, A4.2 and A4.3 as follows:

[0123] For the preparation of the two-component reactive resin systems, the reactive resin components (component (A)) were combined with a hardener component (component (B)) of the commercially available product HIT HY-200 (Hilti Aktiengesellschaft: batch number: 8103926) and filled into plastic cartridges (Ritter GmbH; volume ratio A:B=5:1) having inner diameters of 32.5 mm (component (A)) and 14 mm (component (B)).

[0124] The two-component reactive resin systems B5.1, B5.2 and B5.3 were then prepared from the reactive resin components B4.1, B4.2 and B4.3 as follows:

[0125] For the preparation of the two-component reactive resin systems, the reactive resin components (component (A)) were combined with a hardener component (component (B)) of the commercially available product HIT-CT 1 (Hilti Aktiengesellschaft; batch number 8600465) and filled into plastic cartridges (Ritter GmbH; volume ratio A:B=3:1) having inner diameters of 47 mm (component (A)) and 28 mm (component (B)).

[0126] Evaluation:

[0127] The reactive resins A3.1 to A3.3 and the reactive resins B3.1 to B3.3 were examined for their reactivity. Since the curing system of the reactive resins of the A series produces significantly more initiator radicals than the reactive resins of the B series, a comparison of the two systems is intended to show whether there is a difference depending on the initiator radical amount when using reactive diluents containing itaconate. The question to be answered is therefore whether approximately the same amount of itaconate copolymerizes in the two very different systems. The amount of heat released during the reaction was determined as an indirect approximate but sufficient measure for this purpose. For this purpose, the resins were mixed intensively with gypsum-stabilized dibenzoyl peroxide (Perkadox 20S, AkzoNobel). In the case of the A series reactive resins, 70 g reactive resin was mixed intensively with 30 g Perkadox 20S. In the case of the B series reactive resins, 70 g reactive resin was mixed intensively with 6 g Perkadox 20S. The reactivity period was measured; this is understood to mean the resin reactivity (t.sub.r,25.fwdarw.T.sub.max) of a resin or a resinous composition expressed as the time from the time of addition of an initiator to initialize the cure to the time when the composition has reached the maximum temperature (T.sub.max). The maximum temperature (T.sub.max) was also measured. The measurement was carried out using a conventional device (Geltimer, WKS Informatik). Both the reactive resin and the Perkadox were previously heated to 25 C. in a drying cabinet. The mixture was filled into a test tube after the addition of the initiator, up to a height of 4 cm below the rim, the test tube being kept at a temperature of 25 C. (DIN 16945, DIN EN ISO 9396). A temperature sensor which recorded a temperature-time curve was immediately introduced into the mixture. Table 6 shows the results of these measurements:

TABLE-US-00008 TABLE 6 Results of the temperature measurements Reactive t.sub.r, 25 .fwdarw. T.sub.max T.sub.max resin [mm:SS] [ C.] A3.1 08:25 160 A3.2 07:16 147 A3.3 09:24 141 B3.1 09:37 166 B3.2 08:33 145 B3.3 07:02 142

[0128] The reactivity time of the resins is comparable in the usual range of these measurements. The shape of the curves, from which a possible retardation could be identified, was also the same, which indicates normal curing.

[0129] Both positive references A3.1 and B3.1 with 20 wt. % 1,4-butanediol dimethacrylate show that complete polymerization can produce a temperature increase up to approximately 160 and 166 C., respectively. Both negative references A3.3 and B3.3 with 20 wt. % diethyl malonate show that when 20 wt. % unreactive material is used, the temperature can only increase to approximately 141 and 142 C., respectively. The resins of inventive examples A3.2 and B3.2 exhibit a temperature increase to 147 and 145 C., respectively.

[0130] In order to investigate the effects of the methyl butanediol diitaconate oligomer in comparison to the references, the bond stresses of the two-component reactive resin systems were determined. In order to determine the bond stresses (load values) of the cured fixing compositions, M12 anchor threaded rods were inserted into boreholes in C20/25 concrete having a diameter of 14 mm and a borehole depth of 72 mm, which boreholes were filled with the reactive resin mortar compositions. The bond stresses were determined by centric extension of the anchor threaded rods. In each case, five anchor threaded rods were placed and after 24 hours of curing, the bond stress was determined. The fixing compositions were ejected out of the cartridges via a static mixer (HIT-RE-M mixer; Hilti Aktiengesellschaft) and injected into the boreholes. The following borehole conditions were set to determine the bond stress: the borehole was hammer-drilled in dry concrete and made dust-free by cleaning. Placing, curing and extending the anchor rod take place at room temperature. Table 7 shows the results of these measurements. The composite stresses shown are average values from five measurements.

TABLE-US-00009 TABLE 7 Bond stresses Reactive resin Bond stress system [N/mm.sup.2] A5.1 30.7 A5.2 26.8 A5.3 15.5 B5.1 15.9 B5.2 13.3 B5.3 12.3

[0131] The results show that the reactive resin systems A5.2 and B5.2, which each contain the itaconic acid ester according to the invention, have a bond stress comparable to the positive references A5.1 and B5.1, which contain a fossil reactive diluent. By using the biogenic reactive diluent according to the invention, the curing properties and the bond stress of the resulting reactive resin system are not adversely affected.

BRIEF DESCRIPTION OF THE DRAWINGS

[0132] FIG. 1 shows the results of MALDI investigations for determining the molar mass of the oligomeric itaconic acid ester; Sample F is the oligomer used in the practical examples;

[0133] FIG. 2 is a schematic representation of the evaluation of temperature-time curves;

[0134] FIG. 3 shows the temperature-time curves measured in Example 2.