Plaintiffs charge defendant with massive patent infringement. The patent ("Claim 4," "Devex patent," "Henricks process") which is at the root of this litigation provides as follows:

The process of working ferrous metal which comprises forming on the surface of the metal a phosphate coating and superimposing thereon a fixed film of a composition comprising a solid meltable organic binding material containing distributed there through a solid inorganic compound meltable at a temperature below the melting point of the ferrous metal phosphate of said coating and having a hardness not exceeding 5 on the Mohs' hardness scale, and thereafter deforming the metal.

This process is used in cold forming metal car parts by pressure.

The happy result of (the Devex) process is that phosphate, soap and borax work to lubricate the pressure-forming operation, preventing harmful contact between the metal products and the machinery with which they are formed.... (The Devex process) is especially beneficial because it may be easily cleaned from the metal product following its formation.

Devex Corp. v. General Motors Corp., 494 F. Supp. 1369, 1372 (D.Del.1980). The Devex patent expired in 1969; thus, only damages issues are before us.

This litigation, which began twenty-five years ago, is now in the accounting phase. Defendant and plaintiffs cross-appeal from the judgment of the district court, which was based in part on the report of a Special Master. For the reasons which follow, we will affirm.

The complaint in this case was filed November 13, 1956 in the Northern District of Illinois. After a trial, Judge Robson of that court held on February 1, 1962 (final judgment entered June 29, 1962) that the Devex patent was invalid. The United States Court of Appeals for the Seventh Circuit reversed, Devex Corp. v. General Motors Corp., 321 F.2d 234 (7th Cir. 1963), cert. denied, 375 U.S. 971, 84 S. Ct. 490, 11 L. Ed. 2d 418 (1964), and held that the patent was valid because "the elements which Henricks selected were put together in a new way and achieved a new and unexpected result." 321 F.2d at 236. The court further stated:

There is substantial evidence in the record to prove that a new coaction between the soap, borax and phosphate occurred during the drawing process. Friedberg's tests showed that in the Henricks' process, new compounds are formed; the formation of insoluble organic compounds is inhibited, and the abrasive phosphate is transformed into a glassy amorphous compound having highly effective lubricating properties.

321 F.2d at 237. In other words, the use of a borax and soap combination in conjunction with phosphate resulted in improved lubrication of the metal during the deforming (cold forming by pressure) process and in improved cleanability of the product after its formation. The improved lubrication and cleanability combined to make the process patentable.

The court of appeals remanded the case for further proceedings. At that point, General Motors moved to have the case against it transferred to Delaware, and its motion was granted. Devex Corp. v. General Motors Corp., 146 U.S.P.Q. 346 (N.D.Ill.1965).1

Judge Wright of the United States District Court for the District of Delaware ruled in 1967, in the suit between Devex and General Motors, that summary judgment in favor of plaintiffs on the issue of infringement was inappropriate. Devex Corp. v. General Motors Corp., 263 F. Supp. 17 (D.Del.1967). In 1968, he granted a motion by Devex for permission to amend its complaint to assert the doctrine of equivalents. Devex Corp. v. General Motors Corp., 285 F. Supp. 109 (D.Del.1968).2

After an infringement trial, Judge Wright ruled that the patent had not been infringed by General Motors. 316 F. Supp. 1376 (D.Del.1970). That ruling was reversed by this court. Devex Corp. v. General Motors Corp., 467 F.2d 257 (3d Cir. 1972), cert. denied, 411 U.S. 973, 93 S. Ct. 2145, 36 L. Ed. 2d 696 (1973). We rejected the district court conclusion "that infringement was not proved because Devex failed to establish that General Motors achieved the unexpected results that made the Henricks combination patentable." 467 F.2d at 260. After examining the materials before us, we concluded that the accused practices led to satisfactory lubricity and cleanability, which meant that those practices infringed. We specifically rejected General Motors' contention that, because the chemical reactions involved were arguably different from those taught by the Henricks patent, the General Motors practices did not infringe.

(I)f it is directly determinable that the two lubricants have essentially the same components, are applied in the same way and that the results of their use are essentially the same, the disputation of chemists about the chemical interactions that occurred in the processes cannot be decisive.

Moreover, it may well be that chemical reactions differed in degree, and perhaps in kind as well, in different applications of the Henricks process. Claim 4, even as restricted and held valid in the Seventh Circuit, covers and protects the combination of soap and borax in proportions to be varied according to the nature and difficulty of the draw to be facilitated so that the process will yield its claimed advantages over a wide range of drawing pressures and temperatures. It is not claimed that in every Henricks mix under every temperature, chemical reaction and its products will be identical. Thus, if one accepts as persuasive the testimony of GM experts that reactions and substances produced during the GM drawings were not identical with those that the Henricks process is said to have produced using different proportions of ingredients under different pressures at different temperatures, it does not follow that GM practice failed to produce the results that made the Henricks process patentable.

In these circumstances, the demonstrated effectiveness of GM practice in producing the satisfactory drawing and ease of cleaning that Henricks had achieved must prevail over any contrary inference drawn from disputed expert testimony as to comparative chemistry in what may well not have been equivalent situations.

467 F.2d at 261-62. Thus, because the results produced by the accused practices were identical to those produced by the Devex patent, the accused practices infringed. The differences between the specific chemical processes of the Devex patent and those of the accused practices were not determinative, so long as the results were identical.II. Issues Now on Appeal

(a) Borax-based lubricants

Based upon our prior opinion, 467 F.2d at 261, the Special Master ruled that the accused practices which involved the use of borax-based lubricants (soap and borax) in cold forming bumpers and non-bumpers infringed the Devex patent, where the use of those lubricants led to cleanability in conjunction with lubricity, and where the advantages of both the cleanability and lubricity were used by General Motors. Thus, the Special Master concluded that accused bumper practices 4, 5, 12, 13, 36, 48, 49, 51, and non-bumper practices 7, 9, 16, 17, 22, 25 (in part), 26, 27, 28, 30, 31, 37, 38, 42, 47, 54 and 56 infringed the Devex patent. Special Master's Report at 27; Appendix at 287.

(b) TKPP and TSP substituted for borax in lubricants

The Special Master also ruled that General Motors' use of tetrapotassium pyrophosphate (TKPP) as a borax substitute in bumper making (accused practices 6, 14, 50, 52, and 53) and of trisodium phosphate (TSP) as a borax substitute in non-bumper making (accused practices 3, 11, 15 and 45) infringed the Devex patent. Special Master's Report at 29-38; Appendix at 289-298.

The Special Master found as a fact that "TKPP was put in (to the lubricant) as a borax substitute, to help the soap do the job of efficiently lubricating the workpiece." Special Master's Report at 31; Appendix at 291. General Motors argued that the use of TKPP did not infringe the patent because pure TKPP melts at a higher temperature than the ferrous metal phosphate coating, which meant that TKPP did not fit the literal language of the patent. The Special Master rejected that argument, found as a fact that TKPP, when mixed in the coatings and as a matter of actual practice, melts before the ferrous metal phosphate coating, and concluded that the use of TKPP as a borax substitute to improve lubricity and cleanability therefore infringed the Devex patent. Special Master's Report at 32-37; Appendix at 292-297. The Special Master also found that TSP (accused practices 3, 11, 15, 45) has a lower melting point than the ferrous metal phosphate coating, and that if it were used as a borax substitute to obtain improved lubricity and cleanability it would infringe the Devex patent. Special Master's Report at 37-38; Appendix at 297-98.

(c) Accused practices involving no actual cleaning

The Special Master held that accused practices 2, 3, 10, 11, 15, 19, 20, 21, 25 (in part), 33, 34, 35, 39, 40, 41, 44 and 45 did not infringe the Devex patent. Some of those practices involved little or no actual cleaning of the part after its formation; some involved a cleaning process which, unlike the easy cleaning process used on borax-treated parts, would have removed the residue even if the Devex process had not been used. Because "ease of cleaning is an element vital to the life of the patent," the Special Master held that those practices which do not involve cleaning of the part cannot infringe the patent. Special Master's Report at 39-43; Appendix at 299-303.3

(d) Borax Rinses

Accused Practices 1, 2, 8, 18, 20, 23, 24, 32, 34, 35, 39, 43, 55, 57 and 58 allegedly involved the use of a neutralizing rinse (as opposed to a lubricant) containing borax.4 The Special Master concluded that the use of rinses containing borax did not infringe claim 4, for the following reasons:i) There was no evidence that borax (or equivalent) rinses were used to lubricate the metal or to aid in cleaning. Special Master's Report at 46; Appendix at 306.

ii) The borax was used solely to neutralize acid carryover resulting from the phosphate bath. Numerous other techniques which did not involve borax were also used to achieve this result. Special Master's Report at 46-47; Appendix at 306-07.

iii) Example XXXV in the 1950 Henricks patent application set forth the use of a borax rinse to neutralize the acid. Example XXXV was cancelled by Henricks when the "patent examiner required that the specification be shortened and limited to 'such parts of it as are commensurate with claims that applicant will continue to prosecute.' " Moreover, Henricks testified that "neutralizing rinses were old and well known." Special Master's Report at 47-48; Appendix at 307-08.

iv) Henricks did not invent the use of borax as a neutralizing rinse. Special Master's Report at 48-49; Appendix at 308-09.

v) Defendant thought it was using borax rinses to neutralize acid, and not for lubrication or cleanability. Defendant used non-accused rinses as well as accused rinses, and easily shifted back and forth. Special Master's Report at 49-50; Appendix at 309-10.

vi) Plaintiffs could have, but did not, contend that certain of the borax rinse processes infringed the patent prior to and at the infringement trial. Special Master's Report at 50; Appendix at 310.

vii) It would make no sense for defendant to seek the benefits that the borax-containing lubricant could provide in a rinse, rather than by the use of the borax-containing lubricant. Special Master's Report at 51; Appendix at 311.

(e) Accused Practices 1, 8 and 46

Plaintiffs argued that Accused Practices 1, 8 and 46 involved the use of borax, and that they infringed Claim 4 for that reason.

The Special Master discussed the factual question of whether Accused Practices 1, 8 and 46 contained borax at pages 53-59 of his report (Appendix at 313-319), and, after weighing the evidence adduced by each side, concluded that they did not.

(a) Multiple Damages and Attorney's Fees

The Special Master ruled that an award of multiple damages and of counsel fees would be inappropriate in this case. "(T)wo highly respected jurists (Judges Robson and Wright) have reached the selfsame conclusions for which plaintiffs now accuse defendant of bad faith." Special Master's Report at 67-68; Appendix at 327-28. The Special Master also found explicitly that defendant had not been guilty of bad faith. Special Master's Report at 70; Appendix at 330.

(b) Infringement in Bumper-Making: Facts

§ 284. Damages

Upon finding for the claimant the court shall award the claimant damages adequate to compensate for the infringement, but in no event less than a reasonable royalty for the use made of the invention by the infringer, together with interest and costs as fixed by the court.

The Special Master, noting that plaintiffs were not themselves the manufacturers, ruled that damages would take the form of a reasonable royalty, and that the royalty would be ascertained "by reference to hypothetical negotiations." Special Master's Report at 71; Appendix at 331.

The Special Master first examined the Devex patent as used in bumper making. The Oldsmobile division of General Motors, throughout the life of the Devex patent, cold formed bumpers without infringing the Devex patent. The Special Master thus found as a fact that "Oldsmobile was successful in cold forming bumpers without the use of the Henricks process during the entire life of the patent." Special Master's Report at 77; Appendix at 337. "To sum up, Oldsmobile's experience shows it was commercially practicable throughout the life of the Henricks patent to cold form bumpers without utilizing the Claim 4 process." Special Master's Report at 79; Appendix at 339.5

Bumpers at the Chevrolet division of General Motors were hot forged until 1954. In 1954, a new plant began cold forming bumpers for Chevrolet; the lubricant used was plain soap over phosphate. Bumper production was satisfactory without borax or an equivalent.

By 1957-58, Chevrolet was experiencing plating problems,6 and in November 1958, Chevrolet switched to a lubricant which contained borax and infringed the Devex patent. From November 1958 to mid-January 1965, Chevrolet used infringing processes (containing borax) to manufacture 93% of its bumpers. Chevrolet used non-infringing processes to a limited extent (7%) during that period. From 1965 until the patent expired, Chevrolet used infringing processes (containing TKPP) to cold-form approximately 84% of its bumpers. In sum, the Special Master found that Chevrolet used the Devex process because Chevrolet thought the Devex process was better, but rejected plaintiffs' contention that the Devex process was essential. In reaching this conclusion, the Special Master carefully weighed conflicting testimony. Special Master's Report at 79-91; Appendix at 339-351.

The Pontiac division of General Motors used a plastic coating (non-infringing) process in making its bumpers from 1952-1955. From late 1955 through the 1968 model year, Pontiac used infringing processes to make its bumpers, with the exception of the 1956 model year, during which Pontiac used the plastic spray for some of its bumpers. In August 1968, Pontiac switched to a non-accused process. Special Master's Report at 91-96; Appendix at 351-356.

The Special Master found that the Henricks process in use at Pontiac was superior to non-infringing practices, but that the Henricks process was "not indispensable." Special Master's Report at 97 & n., Appendix at 357 & n. After weighing conflicting evidence, the Special Master concluded that "Henricks' lubricants came to be regarded by Pontiac as more satisfactory, but others would have done the job." Special Master's Report at 98; Appendix at 358.

The Cadillac division of General Motors cold formed its bumpers from 1952 through the 1962 model year without infringing the Devex patent. Cadillac switched to the Devex process in the 1962 model year, and used it until July 1968 for 100% of its bumpers. In 1968, Cadillac switched to a non-infringing practice. Again, the Special Master found that the Henricks process resulted in benefits to General Motors, but that it was not essential. Special Master's Report at 100-01; Appendix at 360-61.

At Fisher Body-Elyria Bumper Parts, General Motors infringed the patent from April 1957 through March 1960. Non-accused or non-infringing practices were used at all other times relevant to this suit. Special Master's Report at 101-02; Appendix at 361-62.

The Special Master's factual finding as to infringement in bumper making, in sum, is that the three major infringing divisions (Pontiac, Chevrolet and Cadillac) used the Henricks process because it was better than non-infringing alternatives, but not because it was essential or because there were no commercially viable non-infringing alternatives available. Indeed, Oldsmobile's experience proves the availability of practicable alternatives. There were periods when one or more of the infringing divisions used non-accused or non-infringing practices. This leads inexorably to the factual conclusion that non-infringing cold forming alternatives to the Devex process existed: if there had been no choice, defendant would have infringed the patent 100% of the time.

(c) The Reasonable Royalty: Bumpers

Plaintiffs sought as a royalty for bumper infringement a percentage of the sale price of the bumpers. The percentage plaintiffs sought depended upon their factual contention that the Devex process was essential to defendant's cold forming of bumpers. The Special Master rejected plaintiffs' premise when he found as a fact that non-infringing, commercially practicable cold forming processes existed. He also rejected the royalty rates proposed by General Motors, which were based in the alternative on the cost of the lubricants, the sale price of the patent, the value given the patent for estate tax purposes, settlements reached by plaintiffs with other infringers, and the royalty rate which the owners of the patent charged a company owned by the same persons who owned the patent at the time.

The Special Master decided to base the reasonable royalty on an offer made by plaintiffs in 1964 to license the patent for three quarters of one percent (.75%) of the sale price of the bumpers. The Special Master postulated hypothetical negotiations, stressed the availability of viable alternatives to the Devex patent, added that Oldsmobile produced bumpers without the Devex process but that Chevrolet, Pontiac and Cadillac were loyal to the Devex process, noted that the Devex process did have a considerable value, and ultimately reduced the .75% figure-which he decided would have been plaintiffs' opening offer-by approximately one-third.

(d) Infringement in Non-Bumper making: Facts and Reasonable Royalty

Plaintiffs argued before the Special Master that defendant's choice was between "cold forming non-bumper parts through Henricks, or not cold forming at all." Special Master's Report at 103; Appendix at 363. The Special Master rejected this contention, and

(found) the fact to be that non-bumper parts could be and very often were, cold formed by defendant without the use of Henricks. Indeed the question does not even seem to me to be a close one. The evidence is that, far from dominating the field, the Henricks process had relatively slight importance to defendant in the production of non-bumper extruded parts.

Special Master's Report at 104; Appendix at 364.

The Special Master then examined the impact of this conclusion on plaintiffs' damages claims for non-bumper infringement. Plaintiffs sought a percentage of the difference between the cost of producing non-bumper parts by matching or hot forging and by using the Devex process. The Special Master noted that a reasonable royalty cannot be based on speculation, although "a fair approximation based on the evidence will suffice." Special Master's Report at 111; Appendix at 371. He continued:

By relying in their savings-and-gains proof solely on the (erroneous, as I have found) premise that defendant could have cold formed non-bumper parts only through Henricks, plaintiffs have offered no evidence on which a royalty might be reasonably based.

Special Master's Report at 111-12; Appendix at 371-72. Although there had been sufficient evidence with regard to bumpers to supply a fall-back position (the .75% offer), there was no such alternative presented with regard to non-bumper parts, and plaintiffs were therefore not entitled to royalties on the non-bumper infringement of the patent.

(e) Interest

The Special Master awarded interest "as a matter of fact," after finding that the hypothetical bargainers would arrange for interest as part of the reasonable royalty. Special Master's Report at 140-41; Appendix at 400-01. He also ruled that plaintiffs were entitled to prejudgment interest (delay damages) at the prevailing corporate bond rate as a matter of law, and that that interest should be calculated each year based on the number of infringing parts made in that year.

The district court, turning first to plaintiffs' claims of error, affirmed the ruling of the Special Master that "(p)ractices as to which little or no cleaning was required do not infringe." 494 F. Supp. at 1375. The trial court interpreted prior decisions in the Devex litigation as "holding that Claim 4 requires cleaning as an integral step of its process." Id. at 1373. The court stated:

The primary importance of Devex's process was not that the addition of borax somehow enhanced lubricity. Rather, the addition of borax inhibited the formation of zinc stearate resulting in products that could be easily cleaned. (In the validity phase of this litigation), Devex ... relied upon the ease of cleaning to defend the validity of what would otherwise have been an overly broad claim.

494 F.Supp. at 1374-75.

The trial court then turned to the issue of whether borax rinses infringed the Devex process. He ruled that the borax rinses did infringe the patent, but only where the parts formed after the rinse was applied were cleaned. In other words, if the rinse performed the same role as the lubricant, it infringed the patent.

The trial court affirmed the Special Master's fact finding that accused practices 1, 8 and 46 did not infringe, noting that the "clearly erroneous" standard of review applied under Fed.R.Civ.P. 53(e)(2), and that the Special Master had "meticulously reviewed and resolved this evidentiary dispute ...." 494 F. Supp. at 1376.

The trial court then turned to the reasonable royalty rate. The court agreed with the Special Master's findings that

... Oldsmobile used non-infringing practices to make almost all of the bumpers. Similarly Chevrolet relied, though less successfully, upon non-infringing processes until November, 1958, and used non-infringing processes to manufacture part of its bumper needs from 1960 to 1963 and from 1965 to 1966. Cadillac used non-infringing processes prior to 1962 and after 1968. Fisher Body used non-infringing processes prior to 1957 and after 1960. Where G.M.'s various divisions so often produced marketable cars without using Devex's process, the Court finds, as the Master did also, that G.M.'s negotiators would not have agreed to the high figure Devex now proposes.

494 F.Supp. at 1376-77. The court disagreed, however, with the reduction in the .75% licensing offer, and ruled that in the absence of evidence tending to show that Devex would accept a figure less than .75%, the reduction was improper because speculative. Thus, the court accepted .75% of the sale price as the royalty for bumper infringement.7

The court affirmed the Special Master's refusal to award royalties for non-bumper infringement:

In this case, the Master painstakingly reviewed the benefits of Devex's process versus those of other non-infringing pressure forming operations. He found that G.M.'s various divisions easily substituted non-infringing processes to manufacture non-bumper parts. Apparently, then, Devex's process was of "relatively slight importance to" G.M., and for this reason the Master properly found that G.M. should not be required to pay anything for its infringement in the manufacture of non-bumper parts.

The trial court then turned to General Motors' claims of error. The court rejected General Motors' contention that this court, in discussing Bonderlube 235 in its prior opinion, intended only to reverse the judgment of non-infringement as to Bonderlube 235 and to leave the rest of the trial court judgment, which dealt with other compounds, intact. The trial court also rejected General Motors' contentions that TKPP and TSP did not infringe, that the royalty rate should have been based on other factors, that the Special Master should not have assessed prejudgment interest, and that General Motors should not be liable for costs (including the Special Master's fee).

Plaintiffs make the following arguments on appeal to this court:

i. The award of a royalty of .75% of bumper sales was unconscionably low. The Special Master and the trial court erred in finding that there were reasonable alternatives to the Devex process available to defendant. Not only were the fact findings of the lower courts clearly erroneous, but also the court and the Special Master erred as a matter of law in disregarding (or refusing to rely upon) plaintiffs' evidence of savings attained through use of the Devex process. The lower court further erred as a matter of law in not relying upon plaintiffs' evidence of savings, because defendant did not refute plaintiffs' showing that there were no alternatives available to General Motors at the time the infringement began which would have yielded results equally beneficial to the results given by the Devex process.

ii. The finding that accused practices 8 and 46 did not contain borax or TKPP was clearly erroneous. Moreover, that defendant admitted that accused practice 46 infringed the patent.

iii. The Special Master and the trial court erred as a matter of fact and of law in refusing to award royalties for non-bumper infringement. The finding that there were alternatives available to General Motors which would give equally beneficial results was clearly erroneous. Moreover, the patent laws require the award of a reasonable royalty where infringement has taken place; thus, to refuse to award damages in a case such as this one, where massive infringement occurred, is erroneous. Finally, General Motors failed to rebut plaintiffs' showing as to what the savings resulting from infringement were, and failed to show that there were in fact alternatives available to General Motors at the time infringement began which would yield equally beneficial results to the Devex process.

iv. The Special Master and the trial court erred in construing the prior opinions written in this case as requiring that a part formed by the Devex process actually be cleaned in order for infringement to have occurred. Attaining the result of cleanability is enough to warrant a finding of infringement, even though the part is not actually cleaned.

Defendant raises the following contentions on this appeal:

i. It is erroneous for a court to award prejudgment interest on a reasonable royalty award where the amount of the award was unliquidated until judgment was entered, and in the absence of special circumstances.

ii. The law of the case has restricted the Devex patent to those processes which involve the use of borax. Thus, plaintiffs cannot now argue that the use of TKPP and TSP infringes the patent, both because of the law of the case and because plaintiffs earlier waived the right to assert the doctrine of equivalents.

iii. The Special Master was correct in ruling that the use of rinses which contain borax or its equivalents does not infringe the patent, because the law of the case requires that the use of the Devex patent be for the purposes of attaining lubricity and cleanability, and the use of the rinses has a different purpose entirely. Thus, the district court erred in ruling that the use of the rinses infringes the Devex patent.

iv. The choice by the Special Master of the .75% royalty rate, and his subsequent reduction of that figure by one-third, was erroneous because the correct figure was one of the alternatives suggested by defendant. The increase by the trial court of the rate to the .75% figure compounded the error.

In patent infringement cases, "(t)he awarding of damages involves a subtle judicial function; the statute allows considerable latitude (to the fact-finding court)." John Zink Co. v. National Airoil Burner Co., 613 F.2d 547, 559 (5th Cir. 1980). See also Milgo Electronic Corp. v. United Business Communications, Inc., 623 F.2d 645, 663 (10th Cir.), cert. denied, 449 U.S. 1066, 101 S. Ct. 794, 66 L. Ed. 2d 610 (1980) ("An appellate court must affirm the award if there is substantial record evidence to support it: the trial court's findings will not be set aside unless it is shown that they are clearly erroneous.")

The scope of a patent, which is determined by construing the patent, is a question of law. Studiengesellschaft Kohle mbH v. Eastman Kodak Co., 616 F.2d 1315, 1324 (5th Cir.), cert. denied, 449 U.S. 1015, 101 S. Ct. 573, 66 L. Ed. 2d 473 (1980). However, "(t)he issue of patent infringement is a question of fact...." Id. at 1322. Moreover, "patent cases, because they so frequently contain conflicts in expert testimony, seem particularly suited for Rule 52(a)' § review limitations (the clearly erroneous standard)." Id. at 1323.

The Special Master heard testimony during fifty-three days of trial. He wrote an exhaustive and thoughtful report, in which he set forth findings of fact based on a careful weighing of conflicting testimony as well as of documentary evidence. The trial judge, after a detailed analysis of the facts found by the Special Master, and after applying the standard of review set forth in Fed.R.Civ.P. 53(e)(2) (the "clearly erroneous" standard) to those fact findings, decided that none of the Special Master's findings of fact was clearly erroneous. We agree.

Whether the Special Master and the trial court ascertained and applied the law correctly to the facts is subject to a less stringent standard of review than the "clearly erroneous" standard. We have concluded, however, that the Special Master and the trial court ascertained and applied the law correctly. The two differ in only two respects: first, the Special Master ruled that none of the accused rinse practices infringed the patent, while the trial court ruled that those accused rinse practices which involve no cleaning do not infringe, but that those which involve cleaning do infringe. Second, the Special Master reduced the .75% industry-wide offer figure by one-third, while the trial court reinstated the .75% figure. We agree with the results reached by the Special Master and the district court; where the two differ, we affirm the district court.

The scope of the Devex patent is largely a question of law. Thus, questions such as whether cleanability (as opposed to actual cleaning) is sufficient to bring a process within the patent are questions of law. The question of which of the accused processes infringed the patent is one of fact. Which processes involved actual cleaning, which processes involved the use of borax equivalents, and the like, are factual issues.

When it comes to review of the damages issues, such as whether plaintiffs proposed a reasonable standard of comparison upon which the court could have based a royalty award, identifying a question as one either of fact or of law becomes more problematic. At this point in this case, however, the reasonableness of a proposed standard of comparison is largely factual, because the answer depends upon factual issues, such as whether noninfringing alternatives were available to defendant. If such alternatives were available, it would weigh heavily against plaintiffs' proposed standard of comparison if that comparison depended upon the nonavailability of viable alternatives.

It is clear from prior opinions in this litigation and we reaffirm that the use of borax-based lubricants in cold forming bumpers and non-bumpers infringed the Devex patent where those lubricants contributed to cleanability and lubricity, and where General Motors took advantage of the lubricity and particularly of the cleanability in the actual manufacturing process. Ascertaining which of the accused practices fits this category then becomes a matter of fact. No effective challenge to the inclusion of accused practices in this category has been made.9

2. TKPP and TSP Substituted for Borax in Lubricants: The Doctrine of Equivalents

Plaintiffs have asserted the doctrine of equivalents with regard to defendant's use of TKPP and TSP as borax substitutes. In Graver Tank & Mfg. Co. v. Linde Air Products Co., 339 U.S. 605, 70 S. Ct. 854, 94 L. Ed. 1097 (1950), the Supreme Court discussed the doctrine of equivalents:

The theory on which (the doctrine) is founded is that "if two devices do the same work in substantially the same way, and accomplish substantially the same result, they are the same, even though they differ in name, form, or shape." (Citation omitted.) The doctrine operates not only in favor of the patentee of a pioneer or primary invention, but also for the patentee of a secondary invention consisting of a combination of old ingredients which produce new and useful results ....

Id. at 608, 70 S.Ct. at 856. The Court also established the standard of review applicable to the doctrine:

A finding of equivalence is a determination of fact .... Like any other issue of fact, final determination requires a balancing of credibility, persuasiveness and weight of evidence. It is to be decided by the trial court and that court's decision, under general principles of appellate review, should not be disturbed unless clearly erroneous. Particularly is this so in a field where so much depends upon familiarity with specific scientific problems and principles not usually contained in the general storehouse of knowledge and experience.

Id. at 609-610, 70 S.Ct. at 857.

"The 'doctrine of equivalents' which governs determinations of infringement requires an assessment of function rather than form in measuring the claims of a patent." N.V. Maatschappij Voor Industriele Waarden v. A.O. Smith Corp., 590 F.2d 415, 421 (2nd Cir. 1978).

A patented process is infringed only by unauthorized performance of substantially the same process steps in substantially the same way to accomplish substantially the same result. It is not necessary that the accused process be practiced with the same or substantially the same apparatus as disclosed by the patentee. (Citation omitted.) Nor is it necessary that the materials used in the process be identical to those disclosed and claimed as long as they are equivalent.

International Glass Co. v. United States, 408 F.2d 395, 400, 187 Ct. Cl. 376 (1969). In International Glass, the accused practices did not infringe because

they do not appropriate the gist of the claimed process; are not a mere variation of the details of the claimed process; and in fact do not substantially follow the mode taught by the patent.

Id. at 401.

"To establish equivalency for the purpose of showing infringement of a patent claim, the patentee has the burden of proving a real identity of means, operation, and result." Ziegler v. Phillips Petroleum Co., 483 F.2d 858, 868 (5th Cir.), cert. denied, 414 U.S. 1079, 94 S. Ct. 597, 38 L. Ed. 2d 485 (1973).

(P)rinciples (of the doctrine of equivalents) are applied to compositions of matter where there is equivalence between chemical ingredients. (Citations omitted.) In (Graver Tank & Mfg. Co. v. Linde Air Products Co., 339 U.S. 605, 70 S. Ct. 854, 94 L. Ed. 1097 (1950)) the Supreme Court spoke to the problem of determining what constitutes equivalency.

"Consideration must be given to the purpose for which an ingredient is used in a patent, the qualities it has when combined with the other ingredients, and the function which it is intended to perform."

Graver, supra, 339 U.S. at 609 (70 S.Ct. at 857) ....

Id. at 869.

In recognition of the fact that a patent would be virtually worthless if it did not protect against devices which incorporate only unimportant variations of the patented device, the doctrine of equivalents provides that a later-developed device will be regarded as the equivalent of the patented device if, although different in form or shape, it "performs substantially the same function in substantially the same way to obtain the same result." (Citation omitted.) In its early development, the doctrine generally was applied in cases involving the equivalence of devices having mechanical components. Today, however, the same principles are applied to compositions where there is equivalence between chemical ingredients. (Citation omitted.)

Chemical Cleaning, Inc. v. Dow Chemical Co., 379 F.2d 294, 296 (5th Cir.), cert. denied, 389 U.S. 1040, 88 S. Ct. 777, 19 L. Ed. 2d 829 (1967). See also Rich Products Corp. v. Mitchell Foods, Inc., 357 F.2d 176, 177, 182 (2d Cir.), cert. denied, 385 U.S. 821, 87 S. Ct. 46, 17 L. Ed. 2d 58 (1966) (defendant infringed patent, where defendant substituted hydroxypropyl methyl, a cellulose, for the alkyl groups specified in the patent, the cellulose was one of the four essential ingredients in the patent, and the substitute chemical performed the same function as the chemical listed in the patent).

Here, the Special Master, in the role of fact finder, found, after weighing the evidence, that TKPP and TSP, when substituted for borax, produce the same results (cleanability and lubricity) that borax produces and fit the literal language of the patent in terms of melting point. Cleanability and lubricity are the results which made the Devex process patentable in the first place; thus, TKPP and TSP, which produce those results and which read on the literal language of the patent, infringe the patent. Certainly the fact findings here, that TKPP and TSP are equivalents of borax and infringe the patent, are not clearly erroneous.10

General Motors contends here that plaintiffs cannot claim the use of TKPP and TSP as infringements because plaintiffs insisted earlier in this litigation that borax infringed, and are therefore estopped from asserting that anything other than borax infringed.

We disagree. We reject General Motors' contention that earlier opinions in this litigation which referred to the use of borax meant the exclusive use of borax and no other inorganic compound, and, for the reasons discussed herein, we affirm the rulings that the use of TKPP and TSP infringed the patent.11

3. Accused Practices Involving No Actual Cleaning

The validity of the Devex patent depended upon the new results of commercially practicable cleanability and enhanced lubricity. However, the precise question of whether cleanability or cleaning infringes the patent was not reached in either the validity or infringement phases. Where cleanability is irrelevant to the creation of the part involved, one of the factors which saved the Devex patent from invalidity is not present, and the patent has not been infringed. In other words, we hold that while one result which saved the patent is cleanability, a practice does not infringe unless it involves actual cleaning.

The Special Master ruled that borax rinses did not infringe the Devex patent, in part because they were rinses used to neutralize acid, and were not designed to improve lubricity or cleanability. The trial court agreed that borax rinses did not infringe where they were used solely to neutralize acid, but disagreed with the Special Master's reasoning insofar as it excluded from infringement those rinse practices which were, in fact, used to promote cleaning.

We agree with the trial court. That the soap and borax (or equivalent) may be applied to metal in a series of steps, rather than in a combined form, or as a rinse rather than a lubricant, is not determinative of whether a particular practice infringes or not. What is determinative is the result produced and the use to which that result is actually put. Where soap and borax are applied to the metal seriatim, instead of in a combined form, the process infringes the Devex patent where cleanability and its effects are put to use in the actual process.12

As did the trial court, we conclude that the fact findings made by the Special Master were not clearly erroneous, and affirm the Special Master and the trial court with regard to Accused Practice 46, which did not infringe the patent.

We now turn to the question of what damages plaintiffs should receive from defendant.

The Special Master concluded, based upon facts found by him, that the Devex process was useful in but not essential to the cold forming of bumpers. We agree that the facts lead to that conclusion.

Oldsmobile only infringed the Devex patent in bumper making in 1962-1963, and produced almost all of its bumpers without infringing the Devex process. From 1958-1965, Chevrolet infringed the Devex patent in producing 93% of its bumpers; from 1965-1969, Chevrolet infringed in producing 84% of its bumpers. Pontiac infringed from 1955-1968; Cadillac did not begin infringing until 1962; and Fisher Body-Elyria only infringed between 1957 and 1960. This means, of course, that General Motors did not infringe at all other times. If plaintiffs' argument that General Motors could not have efficiently cold formed bumpers without the Devex process were correct, then General Motors would have been compelled to use infringing processes in producing 100% of its bumpers. General Motors did not do so, and we reject plaintiffs' argument.

The goal of assessing royalties in patent infringement cases is to place the plaintiff in the same position it would have been in if the patent had not been infringed. The burden is on the party seeking damages to show the amount of those damages. See Reliance Molded Plastics, Inc. v. Jiffy Products, 215 F. Supp. 402, 412 (D.N.J.1963), aff'd per curiam, 337 F.2d 857 (3d Cir. 1964). Where a plaintiff itself uses the patented process in manufacturing, damages for infringement may take the form of lost profits, and the burden is on the plaintiff to show their amount. See Panduit Corp. v. Stahlin Brothers Fibre Works, Inc., 575 F.2d 1152, 1157, 1158 (6th Cir. 1978); Zegers v. Zegers, Inc., 458 F.2d 726, 729 (7th Cir.), cert. denied, 409 U.S. 878, 93 S. Ct. 131, 34 L. Ed. 2d 132 (1972); Universal Athletic Sales Co. v. American Gym, 480 F. Supp. 408, 413 (W.D.Pa.1979). Where, as here, the party alleging infringement does not itself manufacture or use the patented process, compensation may take the form of a reasonable royalty for licensing the use of the patent. As with the manufacturing plaintiff, it is arguable that the licensing plaintiff has the burden of proposing a reasonable royalty. See Wayne-Gossard Corp. v. Moretz Hosiery Mills, Inc., 447 F. Supp. 12, 15 (W.D.N.C.1976), aff'd per curiam, 573 F.2d 191 (4th Cir. 1978). A rule that a nonmanufacturing plaintiff has no obligation to proffer a reasonable basis for calculating a royalty would be unfair to manufacturing plaintiffs and to the courts: the manufacturing plaintiff would have the "heavy burden," Panduit Corp, 575 F.2d at 1158, of proving lost profits, while the non-manufacturing plaintiff would have no obligation to come forward with any evidence at all.

Even if there is no explicit burden on a plaintiff to adduce the basis for a reasonable royalty, it is self-evident that there must be enough evidence in the record to allow the fact finder to ascertain a reasonable royalty. In the absence of any evidence as to what would constitute a reasonable royalty in a given case, a fact finder would have no means of arriving at a reasonable royalty, and none could be awarded.

Plaintiffs argue here that they are entitled to a percentage of the savings which accrued to General Motors by reason of General Motors' infringement of the patent, and that the standard to which the infringing bumpers should be compared (the "standard of comparison") is the cost of producing bumpers by hot forging or by contour polishing after forming. Plaintiffs also contend that the burden shifted to General Motors to show that plaintiffs' proposed standard should not be used.

The Special Master and the trial court agreed that plaintiffs' proposed standard of comparison was not reasonable, because that standard of comparison required accepting the factual premise that General Motors could not efficiently cold form bumpers without the Devex process, a factual premise which the Special Master found to be false. Instead of ruling that plaintiffs had not adduced a reasonable standard of comparison and refusing to award damages on the ground that to do so would involve too much speculation, however, the Special Master awarded damages based upon an industry-wide licensing offer made by plaintiffs in 1964. The district court reversed the Special Master's action in reducing the .75% figure by one-third, but agreed with the Special Master's action in basing the award of royalties on the licensing offer. We agree with that approach, and affirm the award of damages of .75% of the sales price of infringing bumpers.

In Pitcairn v. United States, 547 F.2d 1106, 212 Ct. Cl. 168 (1976), cert. denied, 434 U.S. 1051, 98 S. Ct. 903, 54 L. Ed. 2d 804 (1978), the court awarded the owner of the patent a royalty based on an industry-wide offer to license made by the owner.13 In Pitcairn, the patent owner had agreed with one licensee to a 2% royalty, and had offered to license anyone else at the 2% rate. The court rejected the owner's contentions that the 2% royalty agreement could not be taken into account because it "was mainly the product of compromise to avoid litigation" and that an offer by a patentee is inadmissible to prove value. Id. at 1116-17.

The court further gave the offer and the actual rate "a prima facie title to acceptance as the reasonable royalty ...." Id. at 1118. The court rejected the infringer's contentions that the royalty should be less than 2%:

For the present case, in which there was only one actual licensee during almost the entire span of the infringement years, and this suit began early in the recovery period, the best that we can do is to accept the only royalty rate "offered freely (by the owner) to everyone" in the industry during that period, a rate actually agreed to, for a time, by ... the largest manufacturer.

Id. at 1119.

While there are factors which distinguish this case from Pitcairn, nonetheless Pitcairn supports reliance upon the .75% figure here. That plaintiffs here made their offer in the face of industry-wide infringement14 is somewhat counterbalanced by the fact that plaintiffs, at the time of the offer, had just won a major validity victory in the Seventh Circuit. The .75% offer doubtless included an element of a desire to end this already extensive litigation; that factor, although to be taken into account, does not automatically rule out use of the .75% figure.

In Trio Process Corp. v. L. Goldstein's Sons, Inc., 612 F.2d 1353 (3d Cir.), cert. denied, 449 U.S. 827, 101 S. Ct. 91, 66 L. Ed. 2d 30 (1980), the court stated that "the actual license rate is an important factor in the determination of a reasonable royalty, at least when those royalties prove or tend to prove an established royalty." Id. at 1358. The result reached here-that an actual license offer may, if the facts permit, be a factor in assessing a reasonable royalty-logically follows from Trio Process.

In short, plaintiffs' proposed royalty rate was based on an incorrect factual premise. There was, however, evidence in the record justifying the Special Master's basic reliance on the .75% figure. Accordingly, we rule that the .75% rate assessed by the district court was appropriate here.15

The Special Master and the trial court refused to award plaintiffs damages for infringement in non-bumper cold forming because plaintiffs failed to adduce a reasonable standard of comparison upon which to premise such an award and because the Devex patent was relatively unimportant in non-bumper making. Plaintiffs' proposed standard of comparison suffered from the same infirmities as the proposed standard for bumper making: it was based on a factual assertion (that the parts could not be cold formed without the Devex process) which the Special Master found, as a matter of fact, to be incorrect. We affirm the rejection of plaintiffs' proposed standard of comparison, because the facts foreclose the use of that proposed standard.

Plaintiffs make an additional argument with regard to non-bumpers: the statute provides that the court "shall" award damages in an amount not less than a reasonable royalty, and that the award of no damages does not comply with the statute. We reject this contention. The statute requires the award of a reasonable royalty, but to argue that this requirement exists even in the absence of any evidence from which a court may derive a reasonable royalty goes beyond the possible meaning of the statute. Even if there is no burden of proof on the party seeking damages in this type of case to come forward with a reasonable royalty, there must at the least be enough evidence in the record to allow the factfinder to formulate a royalty. Here, plaintiffs' proposed standard was vitiated by the facts, and there was insufficient evidence in the record upon which the factfinder could rely in formulating an alternative. There was no non-bumper industry-wide offer for the court and Special Master to use in arriving at a reasonable royalty. Thus, plaintiffs are not entitled to collect any royalty for non-bumper infringement.16

The Special Master, affirmed by the district court, awarded interest (delay damages) at the prevailing corporate bond rate on the royalty payments as of the date those payments became due, calculated yearly (number of infringing parts made each year). On appeal, defendant contends that such an award was improper. We disagree, and hold that under the facts of this case the award of interest as the yearly royalty payments became due was not an abuse of discretion.

Title 35 U.S.C. § 284 provides that an award for infringement shall be "in no event less than a reasonable royalty for the use made of the invention by the infringer, together with interest and costs as fixed by the court." The language of the statute does not indicate as of when interest, if awarded, shall be calculated. We are not, however, entirely without guidance on this question. In Aro Manufacturing Co. v. Convertible Top Co., 377 U.S. 476, 84 S. Ct. 1526, 12 L. Ed. 2d 457 (1964), the Supreme Court examined the 1946 amendment to 35 U.S.C. § 284, pursuant to which a claimant became entitled to recover damages instead of damages plus profits:

The purpose of the change was precisely to eliminate the recovery of profits as such and allow recovery of damages only.

"The object of the bill is to make the basis of recovery in patent-infringement suits general damages, that is, any damages the complainant can prove, not less than a reasonable royalty, together with interest from the time infringement occurred, rather than profits and damages." H.R.Rep.No. 1587, 79th Cong., 2nd Sess. (1946), to accompany H.R. 5311, at 1-2; S.Rep.No. 1503, 79th Cong.,2d Sess. (1946), to accompany H.R. 5311 at 2.

377 U.S. at 505-06, 84 S. Ct. at 1542 (footnote omitted). Thus, in Aro the Supreme Court identified the purpose of the amendment, and stated that it was part of that purpose that damages in the amount of a reasonable royalty plus interest "from the time infringement occurred" be recoverable.

Moreover, in a case such as this one, 35 U.S.C. § 284 itself supports the award of interest from the date of infringement. Section 284 requires that the damages be "adequate to compensate for the infringement." The complaint in this case was filed in 1956, and the Special Master awarded damages to plaintiffs for infringement which took place in that year. To do otherwise than award interest from 1956 would give defendant a windfall in the form of the use of the royalty money it should have paid to plaintiffs, and would deprive plaintiffs of money they should have had. Moreover, a failure to award interest as of the date of infringement would encourage defendants to draw out litigation for as long a period as possible. The only sufferer from such a result would be the prevailing plaintiff-the innocent party.17

Nor does the award of interest from the date of infringement require appreciably more effort from those masters or judges with the responsibility of setting the amount due than that which would be expended in any event. Here, the master and trial court labored long and hard to ascertain the extent of the infringement and the amount of royalties due. The award of yearly interest did not add appreciably to this herculean task. The number of bumpers produced by infringing processes per year emerged in this case in connection with establishing exactly which accused practices infringed and the viability of alternatives to infringing practices (which necessitated examining what percentage of the bumpers formed each year were formed with infringing processes). Once the royalty rate was set, the award of interest became relatively simple.18

The award of interest from the year of infringement is consistent with Trio Process, where this court awarded "interest from the date of infringement, and costs" to the owner of the patent. 612 F.2d at 1361. See also Trio Process Corp. v. L. Goldstein's Sons, Inc., 638 F.2d 661, 662 (3d Cir. 1981) (opinion on motion for clarification of mandate). While there was an established license rate in Trio Process, we decline to rule that the absence of such a rate precludes an award of interest on the facts of this case. Especially where, as here, interest from the date of infringement can be readily calculated, if follows from Trio Process and from the purpose of the statute that interest should be calculated from the date of infringement. Policy likewise dictates this result: failure in a case like this one to award interest as of the date of infringement rewards infringers where those infringers have refused to accept or negotiate a royalty, and discourages the amicable licensing of patents.

This result is also consistent with Hartford National Bank & Trust Co. v. E.F. Drew & Co., Inc., 188 F. Supp. 347 (D.Del.1960), aff'd per curiam, 290 F.2d 589 (3d Cir.), cert. denied, 368 U.S. 825, 82 S. Ct. 45, 7 L. Ed. 2d 29 (1961), in which the court, prior to Aro, awarded interest "from the end of each calendar year in which defendant's sales were made." 188 F. Supp. at 349. In addition, see Marvel Specialty Co. v. Bell Hosiery Mills, Inc., 386 F.2d 287 (4th Cir. 1967), cert. denied, 390 U.S. 1030, 88 S. Ct. 1409, 20 L. Ed. 2d 286 (1968), in which the court awarded interest on the royalties from the date those royalties had become payable. The royalties in that case were, however, as in Trio Process, based on an established rate at which the injured party had granted actual licenses.

In Pitcairn v. United States, 547 F.2d at 1120-24, the court awarded delay damages "computed on a calendar year basis with interest starting on January 1 of each year on the royalties accrued during the preceding calendar year." Id. at 1121.19 Such an award was necessary to " 'accomplish complete justice as between plaintiff and the United States.' " Id. at 1120, quoting Waite v. U.S., 282 U.S. 508, 509, 51 S. Ct. 227, 75 L. Ed. 494 (1931). The court in Pitcairn used Moody's Composite Index of Yields on Long Term Corporate Bonds to set the delay damage ("interest") rate on the unpaid royalties. 547 F.2d at 1121.

We see no reason to rule that § 284 does not allow an award of interest, or "delay damages," from the date of infringement in the face of the Pitcairn rule that § 1498 allows such an award. Sections 1498 and 284 mandate doing justice to the party whose patent has been infringed. The failure to award interest to plaintiffs here would penalize the prevailing party and reward the infringer for its wrongdoing, and we decline to reach such an inequitable result, one which, moreover, would be inconsistent with Third Circuit precedent as well as with the applicable statute.

The result reached here is inconsistent with that reached in Wahl v. Carrier Manufacturing Co., 511 F.2d 209 (7th Cir. 1975), where the court ruled that "interest should run from the date damages are liquidated." 511 F.2d at 215. The result reached here is not, however, necessarily inconsistent with that reached in Georgia-Pacific Corp. v. U.S. Plywood Champion Papers Inc., 446 F.2d 295, 301 & n.7 (2d Cir.), cert. denied, 404 U.S. 870, 92 S. Ct. 105, 30 L. Ed. 2d 114 (1971), where the court rejected the argument that Aro's quotation of H.R.Rep.No. 1587 constituted an adoption of the rule that interest should run from the date of infringement. In Georgia-Pacific, the court ruled that the award of interest is discretionary with the trial court, and that the trial court had the discretion to award interest as of the date of the last infringement. The court did not rule, however, that it would be an abuse of discretion to award interest as of the date of infringement. But see Milgo Electronic Corp. v. United Business Communications, Inc., 623 F.2d 645, 667-68 (10th Cir.), cert. denied, 449 U.S. 1066, 101 S. Ct. 794, 66 L. Ed. 2d 610 (1980) (in patent cases the normal rule that interest only begins to run after the damages are liquidated is extended to allow the trial judge the discretion to award interest from the date of last infringement).

For the foregoing reasons we affirm the judgment of the District Court.20